OPERATIONS UPON FRACTIONS

Excerpts

OPERATIONS UPON FRACTIONS
To add fractions, find equivalent fractions having a common denominator, add their numerators, write the sum over the common denominator, and reduce to lowest terms.
OPERATIONS UPON FRACTIONS
a-2ba-a-4ba-2b=a^2-4ab+4b^2a(a-2b)-a^2-4aba(a-2b)=4b^2a(a-2b)
OPERATIONS UPON FRACTIONS
In case the terms of the fraction are polynomials, notice that the change in sign affects every term of the numerator or denominator.
OPERATIONS UPON FRACTIONS
To multiply one fraction by another, multiply the numerators together for the numerator of the product and the denominators for its denominator, and reduce to lowest terms.
OPERATIONS UPON FRACTIONS
To divide one fraction by another, invert the divisor and proceed as in multiplication

Equations

OPERATIONS UPON FRACTIONS
\frac{a}{b}=\frac{-a}{-b}=-\frac{-a}{b}=-\frac{a}{-b}
Changing the sign of both numerator and denominator, or of one of them, leaves the value of a fraction unchanged.
OPERATIONS UPON FRACTIONS
\frac{a-b}{x-y}=\frac{b-a}{y-x}=-\frac{b-a}{x-y}=-\frac{a-b}{y-x}
When both terms of a polynomial fraction are polynomials, reversing the sign of every term in the numerator and in the denominator leaves the fraction's value unchanged.
OPERATIONS UPON FRACTIONS
\frac{a-b+c}{x+y+z}=\frac{b-c-a}{-x-y-z}=-\frac{b-c-a}{x+y+z}=-\frac{a-b+c}{-x-y-z}
For a trinomial numerator and denominator, changing the signs of all terms in either or both gives a fraction equal in value to the original.
OPERATIONS UPON FRACTIONS
\frac{b}{c}\div \frac{x}{y}=\frac{b}{c}\times\frac{y}{x}=\frac{by}{cx}
Dividing one fraction by another gives the same result as multiplying by the reciprocal of the divisor.
OPERATIONS UPON FRACTIONS
\left(\frac{a}{b}\right)^2 = \frac{a}{b} \times \frac{a}{b} = \frac{a^2}{b^2}
The square of a fraction is found by squaring its numerator and its denominator separately.
OPERATIONS UPON FRACTIONS
\frac{b^4}{a^4} - \frac{y^4}{x^4} = \left(\frac{b^2}{a^2} + \frac{y^2}{x^2}\right) \left(\frac{b}{a} + \frac{y}{x}\right) \left( \frac{b}{a} - \frac{y}{x}\right)
The difference of two fourth powers of fractions factors into the product of a sum and two differences, as a difference of squares applied twice.
OPERATIONS UPON FRACTIONS
x^4 + x^2 + \frac{1}{4} = \left(x^2 + \frac{1}{2}\right)^2
The expression x^4 + x^2 + 1/4 is a perfect square, equal to (x^2 + 1/2) squared; note the illustration's heading reads x^4 + x^3 + 1/4, which does not match this displayed equation, so the heading appears to be a misprint.

Problems

Exercise 44

Exercise 44, problem 1
$\displaystyle \frac{2a}{3}+\frac{a}{4}-\frac{5a}{6}$.
Printed answer:
- $\displaystyle \frac{a}{12}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a/12
Exercise 44, problem 10
$\displaystyle \frac{2a^2}{a^2-b^2}-\frac{2a}{a+b}$.
Printed answer:
- $\displaystyle \frac{2 a b}{a^{2} - b^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a*b/(a**2 - b**2)
Exercise 44, problem 11
$\displaystyle \frac{2a-3b}{a-2b}-\frac{2a-b}{a-b}$.
Printed answer:
- $\displaystyle \frac{b^{2}}{(a - 2 b)(a - b)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes b**2/((a - 2*b)*(a - b))
Exercise 44, problem 12
$\displaystyle \frac{x+4}{x+5}-\frac{x+2}{x+3}$.
Printed answer:
- $\displaystyle \frac{2}{(x + 5)(x + 3)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2/((x + 5)*(x + 3))
Exercise 44, problem 13
$\displaystyle \frac{x-4y}{x-2y}-\frac{x^2-4y^2}{x^2+2xy}$.
Printed answer:
- $\displaystyle - \frac{4 y^{2}}{x (x - 2 y)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -4*y**2/(x*(x - 2*y))
Exercise 44, problem 14
$\displaystyle \frac{x-7}{x+2}+\frac{x+4}{x-3}$.
Printed answer:
- $\displaystyle \frac{2 x^{2} - 4 x + 29}{(x + 2)(x - 3)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (2*x**2 - 4*x + 29)/((x + 2)*(x - 3))
Exercise 44, problem 15
$\displaystyle \frac{\left(x+2a\right)^2}{x^3-8a^3}-\frac{1}{x-2a}$.
Printed answer:
- $\displaystyle \frac{2 a x}{x^{3} - 8 a^{3}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a*x/(x**3 - 8*a**3)
Exercise 44, problem 16
$\displaystyle \frac{x^6+x^3y^3+y^6}{x^3+y^3}+\frac{x^6-x^3y^3+y^6}{x^3-y^3}$.
Printed answer:
- $\displaystyle \frac{2 x^{9}}{x^{6} - y^{6}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**9/(x**6 - y**6)
Exercise 44, problem 17
$\displaystyle \frac{m - 3}{m + 2} - \frac{m - 2}{m+3} + \frac{1}{m - 1}$
Printed answer:
- $\displaystyle \frac{m^{2} + 11}{(m - 1)(m + 2)(m + 3)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (m**2 + 11)/((m - 1)*(m + 2)*(m + 3))
Exercise 44, problem 18
$\displaystyle \frac{2 (4a + b)}{15 a^2 - 15 b^2} - \frac{1}{5 (a+b)} - \frac{1}{3a - 3b}$
Printed answer:
- 0.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 0
Exercise 44, problem 19
$\displaystyle \frac{3x^3 - 3x^2 + x - 1}{3x^3 - 3x^2 - x + 1} - \frac{3x^3 + 3x^2 - x - 1}{3x^3 + 3x^2 + x + 1}$
Printed answer:
- $\displaystyle \frac{2(9 x^{4} + 1)}{9 x^{4} - 1}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 2*(9*x**4 + 1)/(9*x**4 - 1)
Exercise 44, problem 2
$\displaystyle \frac{2x}{5}+\frac{1}{3} x-\frac{2x}{3}$.
Printed answer:
- $\displaystyle \frac{x}{15}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x/15
Exercise 44, problem 20
$\displaystyle \frac{1}{a^2 - 7a + 12} - \frac{1}{a^2 - 5a + 6}$
Printed answer:
- $\displaystyle \frac{2}{(a - 2)(a - 3)(a - 4)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2/((a - 2)*(a - 3)*(a - 4))
Exercise 44, problem 21
$\displaystyle \frac{a-1}{a-2} + \frac{5 - 2a}{a^2 - 5a + 6} + \frac{a-2}{a-3}$
Printed answer:
- 2.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2
Exercise 44, problem 22
$\displaystyle \frac{1}{a^2 + 3a + 2} + \frac{2a}{a^2 + 4a + 3} + \frac{1}{a^2 + 5a + 6}$
Printed answer:
- $\displaystyle \frac{2}{a + 3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2/(a + 3)
Exercise 44, problem 23
$\displaystyle \frac{x+2}{x-5} + \frac{x-3}{x+4} - \frac{2x^2 - 3x - 2}{x^2 - x - 20}$
Printed answer:
- $\displaystyle \frac{x + 25}{x^{2} - x - 20}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x + 25)/(x**2 - x - 20)
Exercise 44, problem 24
$\displaystyle \frac{m - n}{mn} + \frac{p - m}{mp} + \frac{n-p}{np}$
Printed answer:
- 0.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 0
Exercise 44, problem 25
$\displaystyle \frac{a+b}{ab} - \frac{a-c}{ac} - \frac{c-b}{bc}$
Printed answer:
- $\displaystyle \frac{2}{a}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2/a
Exercise 44, problem 26
$\displaystyle \frac{x^2 - yz}{(x+y)(x+z)} + \frac{x^2 - xz}{(y+z)(y+x)} + \frac{z^2}{(z+x)(z+y)}$
Printed answer:
- $\displaystyle \frac{x^{2} + x z - y z}{(x + z)(y + z)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2 + x*z - y*z)/((x + z)*(y + z))
Exercise 44, problem 27
$\displaystyle \frac{m^2 - bx}{(m+b)(m+x)} - \frac{mx - b^2}{(b+x)(b+m)} - \frac{mb - x^2}{(x+m)(x+b)}$
Printed answer:
- 0.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 0
Exercise 44, problem 28
$x$ is how many times $y$?
Printed answer:
- $\displaystyle \frac{x}{y}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x/y
Exercise 44, problem 29
If $a$ is $\frac{2}{5}$ of a number, what is $\frac{1}{5}$ of the number?
Printed answer:
- $\displaystyle \frac{a}{2}$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem {R: a/2}
Exercise 44, problem 3
$\displaystyle \frac{2x}{3}-\frac{3x}{4}-\frac{4x}{5}$.
Printed answer:
- $\displaystyle \frac{43 x}{60}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 43*x/60
Exercise 44, problem 30
If $x$ is $\frac{3}{7}$ of a number, what is the number?
Printed answer:
- $\displaystyle \frac{7 x}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {N: 7*x/3}
Exercise 44, problem 31
Seven years ago $A$ was four times as old as $B$. If $B$ is $x$ years old, what is $A$’s present age?
Printed answer:
- $4 x - 21$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {A: 4*x - 21}
Exercise 44, problem 4
$\displaystyle \frac{4a}{b}+\frac{3x}{2m}$.
Printed answer:
- $\displaystyle \frac{4 a m + 3 b x}{2 b m}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem (4*a*m + 3*b*x)/(2*b*m)
Exercise 44, problem 5
$\displaystyle \frac{2x}{3}-x+\frac{3x}{5}$.
Printed answer:
- $\displaystyle \frac{4 x}{15}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x/15
Exercise 44, problem 6
$\displaystyle \frac{m^2}{n^2}-2+\frac{n^2}{m^2}$.
Printed answer:
- $\displaystyle \frac{m^{4} - 2 m^{2} n^{2} + n^{4}}{m^{2} n^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (m**4 - 2*m**2*n**2 + n**4)/(m**2*n**2)
Exercise 44, problem 7
$\displaystyle \frac{x}{mn}-\frac{x+mn}{3n}+2m$.
Printed answer:
- $\displaystyle \frac{3 x - m x + 5 m^{2} n}{3 m n}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (3*x - m*x + 5*m**2*n)/(3*m*n)
Exercise 44, problem 8
$\displaystyle \frac{2b+x}{3x}+\frac{5b-4x}{9x}$.
Printed answer:
- $\displaystyle \frac{11 b - x}{9 x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (11*b - x)/(9*x)
Exercise 44, problem 9
$\displaystyle \frac{3m-a}{am}+\frac{b-2m}{bm}-\frac{3b-2a}{ab}$.
Printed answer:
- 0.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 0

Exercise 45

Exercise 45, problem 1
$\displaystyle \frac{2x}{x^2-4} + \frac{1}{2-x} + \frac{1}{2+x}$.
Printed answer:
- $\displaystyle \frac{2}{x + 2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2/(x + 2)
Exercise 45, problem 10
$\displaystyle \frac{1}{(z-a)(z-y)}+\frac{1}{(a-z)(a-y)}+\frac{1}{(y-z)(y-a)}$.
Printed answer:
- 0.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 0
Exercise 45, problem 11
$\displaystyle \frac{1}{(1-x)(1-y)}-\frac{x^2}{(x-1)(y-x)}-\frac{y^2}{(y-1)(x-y)}$.
Printed answer:
- 1.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1
Exercise 45, problem 12
$\displaystyle a-\frac{a^2}{a+1}-\frac{a}{1-a}$.
Printed answer:
- $\displaystyle \frac{2 a^{2}}{a^{2} - 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a**2/(a**2 - 1)
Exercise 45, problem 13
$\displaystyle 1-a+a^2-a^3+\frac{a}{1+a}$.
Printed answer:
- $\displaystyle \frac{1 + a - a^{4}}{1 + a}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (1 + a - a**4)/(1 + a)
Exercise 45, problem 14
$\displaystyle \frac{1}{x^2-7x+12}-\frac{2}{x^2-6x+8}+\frac{2}{x^2-5x+6}$.
Printed answer:
- $\displaystyle \frac{1}{(x - 2)(x - 3)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1/((x - 2)*(x - 3))
Exercise 45, problem 15
What will $a$ pounds of rice cost if $b$ pounds costs 43 cents?
Printed answer:
- $\displaystyle \frac{43 a}{b}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {C: 43*a/b}
Exercise 45, problem 16
$x$ is $\frac{4}{9}$ of what number?
Printed answer:
- $\displaystyle \frac{9 x}{4}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {N: 9*x/4}
Exercise 45, problem 17
$m$ is $\frac{5}{x}$ of what number?
Printed answer:
- $\displaystyle \frac{m x}{5}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {N: m*x/5}
Exercise 45, problem 18
$y$ is $\frac{a}{b}$ of what number?
Printed answer:
- $\displaystyle \frac{b y}{a}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {N: b*y/a}
Exercise 45, problem 2
$\displaystyle \frac{3a}{a^2-9} + \frac{1}{3-a} - \frac{1}{3+a}$.
Printed answer:
- $\displaystyle \frac{a}{a^{2} - 9}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a/(a**2 - 9)
Exercise 45, problem 3
$\displaystyle m^2 - \frac{am^2}{a-m} - \frac{am^2}{m+a} - \frac{2a^2m^2}{m^2-a^2}$.
Printed answer:
- $m^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m**2
Exercise 45, problem 4
$\displaystyle \frac{20a-4}{4a^2-1}+\frac{3}{1-2a}-\frac{6}{1-2a}$.
Printed answer:
- $\displaystyle \frac{1}{2 a + 1}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 1/(2*a + 1)
Exercise 45, problem 5
$\displaystyle \frac{1}{x-y} + \frac{3y^3}{y^3-x^3} - \frac{1}{3(1+a)}$.
Printed answer:
- $\displaystyle \frac{3 y}{x^{2} + x y + y^{2}}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 3*y/(x**2 + x*y + y**2)
Exercise 45, problem 6
$\displaystyle \frac{a+3}{6(a^2-1)}+\frac{1}{2(1-a)}+\frac{1}{3(1+a)}$.
Printed answer:
- $\displaystyle - \frac{1}{3(a^{2} - 1)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -1/(3*(a**2 - 1))
Exercise 45, problem 7
$\displaystyle \frac{1}{5(3b-y^2)}-\frac{1}{2(3b+y^2)}-\frac{11b-7y^2}{10(y^4-9b^2)}$.
Printed answer:
- $\displaystyle - \frac{b}{5(y^{4} - 9 b^{2})}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -b/(5*(y**4 - 9*b**2))
Exercise 45, problem 8
$\displaystyle \frac{3}{(1-x)(3-x)}+\frac{1}{(2-x)(x-3)}-\frac{1}{(x-1)(x-2)}$.
Printed answer:
- $\displaystyle \frac{1}{(1 - x)(3 - x)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1/((1 - x)*(3 - x))
Exercise 45, problem 9
$\displaystyle \frac{a}{(a-b)(b-c)}+\frac{1}{c-a}-\frac{a-b}{(c-a)(c-b)}$.
Printed answer:
- $\displaystyle \frac{b}{(b - c)(a - b)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes b/((b - c)*(a - b))

Exercise 46

Exercise 46, problem 1
$\displaystyle \frac{a}{b^2}$ by $x$.
Printed answer:
- $\displaystyle \frac{ax}{b^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*x/b**2
Exercise 46, problem 10
$\displaystyle \frac{a^2-4a-21}{a^2-3a-10}$ by $a-5$.
Printed answer:
- $\displaystyle \frac{a^2-4a-21}{a+2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a**2-4*a-21)/(a+2)
Exercise 46, problem 11
$\displaystyle x+y-\frac{4xy}{x+y}$ by $x+y$.
Printed answer:
- $x^2-2xy+y^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2-2*x*y+y**2
Exercise 46, problem 12
$\displaystyle a-b+\frac{4ab}{a-b}$ by $a-b$.
Printed answer:
- $a^2+2ab+b^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2+2*a*b+b**2
Exercise 46, problem 13
$\displaystyle \frac{1}{x-y+z}+\frac{2z}{(x-y)^2-z^2}$ by $x^2-xy-xz$.
Printed answer:
- $x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x
Exercise 46, problem 14
$\displaystyle \frac{1}{a-b+c}-\frac{2b}{(a+c)^2-b^2}$ by $ac+bc+c^2$.
Printed answer:
- $c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes c
Exercise 46, problem 15
How many units of the value of $\frac{1}{5}$ are there in 2?
Printed answer:
- $10$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 10
On the STU-32 (STU, rpn):
```
2 ENTER 1 ENTER 5 ÷ ÷
```
Calculator: +1E+1; the book prints 10. Run on the calculator core at firmware 628c96c.
Exercise 46, problem 16
How many units of the value of $\frac{1}{x}$ are there in 5?
Printed answer:
- $5x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*x
Exercise 46, problem 17
How many sixths are there in $4a$?
Printed answer:
- $24a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 24*a
Exercise 46, problem 2
$\displaystyle \frac{3a^2bc}{7x^2y^3}$ by $y^2$.
Printed answer:
- $\displaystyle \frac{3a^2 bc}{7x^2 y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*a**2*b*c/(7*x**2*y)
Exercise 46, problem 3
$\displaystyle \frac{1}{2c^2d}$ by $a^2d$.
Printed answer:
- $\displaystyle \frac{a^2}{2c^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2/(2*c**2)
Exercise 46, problem 4
$\displaystyle \frac{4abc^2}{9x^2y}$ by $3xy$.
Printed answer:
- $\displaystyle \frac{4abc^2}{3x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a*b*c**2/(3*x)
Exercise 46, problem 5
$\displaystyle \frac{3mn}{2x^2y^2}$ by $6x^2y$.
Printed answer:
- $\displaystyle \frac{9mn}{y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*m*n/y
Exercise 46, problem 6
$\displaystyle \frac{x-y}{2mn}$ by $3x$.
Printed answer:
- $\displaystyle \frac{3x(x-y)}{2mn}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x*(x-y)/(2*m*n)
Exercise 46, problem 7
$\displaystyle \frac{2a-b}{3x^2-xy}$ by $x$.
Printed answer:
- $\displaystyle \frac{2a-b}{3x-y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (2*a-b)/(3*x-y)
Exercise 46, problem 8
$\displaystyle \frac{x^2}{x^3-y^3}$ by $x-y$.
Printed answer:
- $\displaystyle \frac{x^2}{x^2+xy+y^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2/(x**2+x*y+y**2)
Exercise 46, problem 9
$\displaystyle \frac{3x+3y}{x-y}$ by $x^2-y^2$.
Printed answer:
- $3\left(x+y\right)^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*(x+y)**2

Exercise 47

Exercise 47, problem 1
$\displaystyle \frac{a^2x}{b}$ by $ax$.
Printed answer:
- $\displaystyle \frac{a}{b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a/b
Exercise 47, problem 10
$\displaystyle \frac{x^2+5x-14}{x^2-7x+12}$ by $x-2$.
Printed answer:
- $\displaystyle \frac{x+7}{x^2-7x+12}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x+7)/(x**2-7*x+12)
Exercise 47, problem 11
$\displaystyle ax+bx+a+b+\frac{a+b}{3x}$ by $a+b$.
Printed answer:
- $\displaystyle x+1+\frac{1}{3x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x+1+1/(3*x)
Exercise 47, problem 12
Multiply $\displaystyle \frac{x^2-5x+6}{x^2+3x-4}$ by $x-1$.
Printed answer:
- $\displaystyle \frac{x^2-5x+6}{x+4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2-5*x+6)/(x+4)
Exercise 47, problem 13
Divide $\displaystyle \frac {5 b c^2}{6 a x^3} $ by $10 a b c$.
Printed answer:
- $\displaystyle \frac{c}{12a^2 x^3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes c/(12*a**2*x**3)
Exercise 47, problem 14
Multiply $\displaystyle 1 + \frac{3ac}{4 x^2 y} $ by $ 2 a c $.
Printed answer:
- $\displaystyle 2ac+\frac{3a^2 c^2}{2x^2 y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a*c+3*a**2*c**2/(2*x**2*y)
Exercise 47, problem 15
Divide $\displaystyle \frac{5 x^2 y - 5 y^3}{2 x^2 z} $ by $ 4 x + 4 y $.
Printed answer:
- $\displaystyle \frac{5y(x-y)}{8x^2 z}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*y*(x-y)/(8*x**2*z)
Exercise 47, problem 16
Divide $\displaystyle \frac{4 x^2 y}{3 m n^2} $ by $ 6 a x y $.
Printed answer:
- $\displaystyle \frac{2x}{9amn^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x/(9*a*m*n**2)
Exercise 47, problem 17
A can do a piece of-work in $2 \frac{1}{3}$ days, and B in $2\frac{1}{2}$ days. How much of the work can they both together do in one day?
Printed answer:
- $\displaystyle \frac{29}{35}$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 29/35
Exercise 47, problem 18
If C can do a piece of work in $m$ days, and D works half as fast as G, how much of the work can D do in one day?
Printed answer:
- $\displaystyle \frac{1}{2m}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 1/(2*m)
Exercise 47, problem 2
$\displaystyle \frac{3m^2n}{2xy}$ by $2x$.
Printed answer:
- $\displaystyle \frac{3m^2 n}{4x^2 y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*m**2*n/(4*x**2*y)
Exercise 47, problem 3
$\displaystyle \frac{18cd^5}{5x}$ by $6cd^3$.
Printed answer:
- $\displaystyle \frac{3d^2}{5x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*d**2/(5*x)
Exercise 47, problem 4
$\displaystyle \frac{6x^2y}{m}$ by $9mn$.
Printed answer:
- $\displaystyle \frac{2x^2 y}{3m^2 n}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**2*y/(3*m**2*n)
Exercise 47, problem 5
$\displaystyle \frac{a}{c+d}$ by $3x$.
Printed answer:
- $\displaystyle \frac{a}{3x(c+d)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a/(3*x*(c+d))
Exercise 47, problem 6
$\displaystyle \frac{3x^3-3xy}{2m^2n}$ by $3x$.
Printed answer:
- $\displaystyle \frac{x^2-y}{2m^2n}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2-y)/(2*m**2*n)
Exercise 47, problem 7
$\displaystyle \frac{2ab-2b^4}{5x^2yz}$ by $2ab$.
Printed answer:
- $\displaystyle \frac{a-b^3}{5ax^2yz}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a-b**3)/(5*a*x**2*y*z)
Exercise 47, problem 8
$\displaystyle \frac{5(a^2-b^2)}{xy}$ by $a-b$.
Printed answer:
- $\displaystyle \frac{5(a+b)}{xy}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*(a+b)/(x*y)
Exercise 47, problem 9
$\displaystyle \frac{m+n}{2m^2-2mn}$ by $m^2-n^2$.
Printed answer:
- $\displaystyle \frac{1}{2m\left(m-n\right)^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1/(2*m*(m-n)**2)

Exercise 48

Exercise 48, problem 1
$\displaystyle \frac{7a^2b^2c}{18x^2y^2z} \times \frac{9xyz^2}{28abc^2}$
Printed answer:
- $\displaystyle \frac{abz}{8cxy}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*b*z/(8*c*x*y)
Exercise 48, problem 10
$\displaystyle \frac{m^2 - 25}{m^2 + 2m - 15} \times \frac{m^4 - 27m}{m^2 - 5m}$
Printed answer:
- $m^2+3m+9$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m**2 + 3*m + 9
Exercise 48, problem 11
$\displaystyle \frac{a^3 - a^2y + ay^2}{x - 3} \times \frac{ax + xy - 3a -3y}{a^3 + y^3}$
Printed answer:
- $a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a
Exercise 48, problem 12
$\displaystyle \frac{x^3 + x^2 - x - 1}{x^2 + 6x + 9} \times \frac{x^2 + 2x - 3}{2x^3 + 4x^2 + 2x}$
Printed answer:
- $\displaystyle \frac{\left(x-1\right)^2}{2x(x+3)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x - 1)**2/(2*x*(x + 3))
Exercise 48, problem 13
$\displaystyle \frac{a^3 - a^2 - a + 1}{a^2 + 4a + 4} \times \frac{a^2 + 3a + 2}{3a^3 - 6a^2 + 3a}$
Printed answer:
- $\displaystyle \frac{\left(a+1\right)^2}{3a(a+2)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a + 1)**2/(3*a*(a + 2))
Exercise 48, problem 14
$\displaystyle (m + \frac{mn}{m - n})(n - \frac{mn}{m + n})$
Printed answer:
- $\displaystyle \frac{m^2 n^2}{m^2-n^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m**2*n**2/(m**2 - n**2)
Exercise 48, problem 15
A grocer purchased $y$ pounds of tea for $25. Another grocer purchased 4 pounds less for the same money. What was the price per pound which each paid?
Printed answer:
- $\displaystyle \frac{25}{y}$; $\displaystyle \frac{25}{y-4}$ dols.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {p: 25/y, q: 25/(y - 4)}
Exercise 48, problem 16
What is $\displaystyle \frac{a}{3}$ of $\displaystyle \frac{2}{c}$?
Printed answer:
- $\displaystyle \frac{2a}{3c}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a/(3*c)
Exercise 48, problem 17
Two numbers differ by 28, and one is eight-ninths of the other. What are the numbers?
Printed answer:
- $252$; $224$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 252, y: 224}
Exercise 48, problem 2
$\displaystyle \frac{9a^2b^2}{8nx^3y^2} \times \frac{5x^2y^2}{2am^2n} \times \frac{24m^2n^2x}{90ab^2}$
Printed answer:
- $ \frac{2}{3}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 2/3
Exercise 48, problem 3
$\displaystyle \frac{x^2 - 16y^2}{xy - 4y^2} \times \frac{2y}{x + 4y}$
Printed answer:
- $2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2
Exercise 48, problem 4
$\displaystyle \frac{3a - b}{2b} \times \frac{18ab + 6b^2}{9a^2 - b^2}$
Printed answer:
- $3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3
Exercise 48, problem 5
$\displaystyle \frac{abd + cd^2}{a^2d - abc} \times \frac{acd - bc^2}{ab^2 + bcd}$
Printed answer:
- $\displaystyle \frac{cd}{ab}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes c*d/(a*b)
Exercise 48, problem 6
$\displaystyle \frac{2ax - 4y}{mn^2 + any} \times \frac{mny + ay^2}{a^2x - 2ay}$
Printed answer:
- $\displaystyle \frac{2y}{an}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*y/(a*n)
Exercise 48, problem 7
$\displaystyle \frac{x^2 - x - 6}{x^2 + x - 2} \times \frac{x^2 + 3x - 4}{x^2 - 2x - 3}$
Printed answer:
- $\displaystyle \frac{x+4}{x+1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x + 4)/(x + 1)
Exercise 48, problem 8
$\displaystyle \frac{a^2 + 2a - 3}{a^2 + a - 2} \times \frac{a^2 + 7a + 10}{a + 3}$
Printed answer:
- $a+5$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + 5
Exercise 48, problem 9
$\displaystyle \frac{x^4 - 64x}{3x^2 - 2x} \times \frac{9x^2 - 4}{x^2 - 4x}$
Printed answer:
- $\displaystyle \frac{(x^2+4x+16)(3x+2)}{x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2 + 4*x + 16)*(3*x + 2)/x

Exercise 49

Exercise 49, problem 1
$\displaystyle \frac{18a^6b^2c}{35x^3y^3z}\div \frac{3a^4b}{5x^2yz}$.
Printed answer:
- $\displaystyle \frac{6a^2bc}{7xy^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 6*a**2*b*c/(7*x*y**2)
Exercise 49, problem 10
$\displaystyle \frac{x^2-y^2}{x^2+y^2}\times\frac{x^4-y^4}{(x-y)^3}\div \frac{(x+y)^2}{x-y}$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1
Exercise 49, problem 11
$\displaystyle \frac{a^2-2a-8}{a^2+2a-15}\div \frac{a^2+5a+6}{a^2-4a+3}\div \frac{a^2-5a+4}{a^2+8a+15}$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1
Exercise 49, problem 12
$\displaystyle \frac{x^2-9x+20}{x^2-5x-14}\div \frac{x^2-7x+12}{x^2-x-42}\times\frac{x^2-x-6}{x^2+11x+30}$.
Printed answer:
- $\displaystyle \frac{x-5}{x+5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x-5)/(x+5)
Exercise 49, problem 13
$\displaystyle \frac{a^2 - a - 2}{a^2 + 2a - 8} \times \frac{a^2 + 5a}{ab + b} \div \frac{a^2 - 25}{a^2 - a - 20}$.
Printed answer:
- $\displaystyle \frac{a}{b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a/b
Exercise 49, problem 14
$\displaystyle \frac{m - 2}{2m - 14} \times \frac{m^2 - 5m - 14}{2m^2 + 4m} \times \frac{4m^2}{m^2 - 4} \div \frac{2mn - 14n}{m^2 - 5m - 14}$.
Printed answer:
- $\displaystyle \frac{m}{2n}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m/(2*n)
Exercise 49, problem 15
$ \displaystyle \left(\frac{a+1}{a-1} + 1\right) \left(\frac{a-1}{a+1} + 1\right) \div \left(\frac{a+1}{a-1} - 1\right) \left(1 - \frac{a-1}{a+1}\right)$.
Printed answer:
- $a^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2
Exercise 49, problem 16
$\displaystyle \left(1 + \frac{x}{1-x}\right) \left(1 - \frac{x}{1+x}\right) \div \left(1 + \frac{1+x}{1-x}\right) \left(1 - \frac{1-x}{1+x}\right)$.
Printed answer:
- $\displaystyle \frac{1}{4x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1/(4*x)
Exercise 49, problem 17
How much times 7 is 21?
Printed answer:
- $3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 3}
On the STU-32 (STU, rpn):
```
21 ENTER 7 ÷
```
Calculator: +3E+0; the book prints 3. Run on the calculator core at firmware 628c96c.
Exercise 49, problem 18
How much times $\frac{3}{14}$ is $\frac{6}{7}$?
Printed answer:
- $4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 4}
On the STU-32 (STU, rpn):
```
6 ENTER 14 ×
```
```
7 ÷
```
```
3 ÷
```
Calculator: +4E+0; the book prints 4. Run on the calculator core at firmware 628c96c.
Exercise 49, problem 19
How much times $\displaystyle \frac{c}{d}$ is $\displaystyle \frac{x}{4}$?
Printed answer:
- $\displaystyle \frac{dx}{4c}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {y: d*x/(4*c)}
Exercise 49, problem 2
$\displaystyle \frac{7axy^2}{8m^2n}\div \frac{2xz^3}{3ab^2m}$.
Printed answer:
- $\displaystyle \frac{21ab^2y^2}{16mnz^3}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 21*a*b**2*y**2/(16*m*n*z**3)
Exercise 49, problem 20
How much times $\displaystyle \frac{5bz}{7mx^3}$ is $\displaystyle \frac{3xy^2z}{4amn^2}$?
Printed answer:
- $\displaystyle \frac{21x^4y^2}{20abn^2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {w: 21*x**4*y**2/(20*a*b*n**2)}
Exercise 49, problem 3
$\displaystyle \frac{4x^2y^3z^4}{3a^2b^3c^4}\div \frac{3x^4y^3z^2}{4a^4b^3c^2}$.
Printed answer:
- $\displaystyle \frac{16a^2z^2}{9c^2x^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 16*a**2*z**2/(9*c**2*x**2)
Exercise 49, problem 4
$\displaystyle \frac{2am^3n^2}{3x^3yz^2}\div \frac{3a^3mn^2}{2x^3yz^2}$.
Printed answer:
- $\displaystyle \frac{4m^2y^2}{9a^2x^2}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 4*m**2*y**2/(9*a**2*x**2)
Exercise 49, problem 5
$\displaystyle \frac{x^2-7x+10}{x^2}\div \frac{x^2-4}{x^2+5x}$.
Printed answer:
- $\displaystyle \frac{x^2-25}{x^2+2x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2-25)/(x**2+2*x)
Exercise 49, problem 6
$\displaystyle \frac{x-6}{x-3}\div \frac{x^2-3x}{x^2-5x}$.
Printed answer:
- $\displaystyle \frac{(x-5)(x-6)}{(x-3)(x-3)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x-5)*(x-6)/((x-3)*(x-3))
Exercise 49, problem 7
$\displaystyle \frac{x-2}{x-7}\div \frac{x^2-9x+20}{x^2-10x+21}\div \frac{x^2-4x+3}{x^2-5x+4}$.
Printed answer:
- $\displaystyle \frac{x-2}{x-5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x-2)/(x-5)
Exercise 49, problem 8
$\displaystyle \frac{a^2-7a}{a-8}\div \frac{a^2+10a+24}{a^2-14a+48}\div \frac{a^2-7a+6}{a^2+3a-4}$.
Printed answer:
- $\displaystyle \frac{a(a-7)}{a+6}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*(a-7)/(a+6)
Exercise 49, problem 9
$\displaystyle \frac{a+b}{(a-b)^2}div \frac{a^2+b^2}{a^2-b^2}\times\frac{a^4-b^4}{(a+b)^3}$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1

Exercise 50

Exercise 50, problem 1
$\displaystyle \left(\frac{2x^2}{ab^3}\right)^2$.
Printed answer:
- $\displaystyle \frac{4x^4}{a^2b^6}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**4/(a**2*b**6)
Exercise 50, problem 10
$\displaystyle \sqrt{ \frac{ 36m^2n^8 }{ 121a^4b^6 } }$.
Printed answer:
- $\displaystyle \frac{6mn^4}{11a^2b^3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 6*m*n**4/(11*a**2*b**3)
Exercise 50, problem 11
$\displaystyle \sqrt[4]{ \frac{ 256x^4y^8 }{ 81a^4b^{12} } }$.
Printed answer:
- $\displaystyle \frac{4xy^2}{3ab^3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x*y**2/(3*a*b**3)
Exercise 50, problem 12
$\displaystyle \sqrt[5]{ - \frac{32x^5y^10}{ 243m^10n^15 } }$.
Printed answer:
- $\displaystyle -\frac{2xy^2}{3m^2n^3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -2*x*y**2/(3*m**2*n**3)
Exercise 50, problem 13
$\displaystyle \sqrt[3]{ -\frac{ 27x^6(a-b)^9 }{ 64a^6b^{12} } }$.
Printed answer:
- $\displaystyle -\frac{3x^2\left(a-b\right)^3}{4a^2b^4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -3*x**2*(a-b)**3/(4*a**2*b**4)
Exercise 50, problem 14
$\displaystyle \sqrt{ \frac{ 4x^2 + 12xy + 9y^2 }{ 16x^4y^8 } }$.
Printed answer:
- $\displaystyle \frac{2x+3y}{4x^2y^4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (2*x+3*y)/(4*x**2*y**4)
Exercise 50, problem 15
$\displaystyle \sqrt{ \frac{ 8x^2(x+y)^5 }{ 50y^4(x+y) } }$.
Printed answer:
- $\displaystyle \frac{2x\left(x+y\right)^2}{5y^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x*(x+y)**2/(5*y**2)
Exercise 50, problem 16
$\displaystyle \sqrt[3]{ -\frac{ 81a^3(a-b)^7 }{ 24b^9(a-b) } }$.
Printed answer:
- $\displaystyle -\frac{3a \left(a-b \right)^2}{2b^3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -3*a*(a-b)**2/(2*b**3)
Exercise 50, problem 17
$\displaystyle \left( \frac{x}{y} + \frac{a}{b} \right) \left( \frac{x}{y} - \frac{a}{b} \right)$.
Printed answer:
- $\displaystyle \frac{x^2}{y^2}-\frac{a^2}{b^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2/y**2 - a**2/b**2
Exercise 50, problem 18
$\displaystyle \left(\frac{a}{b} - \frac{c}{d} \right) \left( \frac{a}{b} + \frac{c}{d} \right)$.
Printed answer:
- $\displaystyle \frac{a^2}{b^2}-\frac{c^2}{d^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2/b**2 - c**2/d**2
Exercise 50, problem 19
$\displaystyle 3ab\left(\frac{2x}{ab}+ \frac{x^2}{2mn}- \frac{ax}{3mb}\right)$.
Printed answer:
- $6x+\frac{3abx^2y}{2mn}-\frac{a^2x}{m}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 6*x + 3*a*b*x**2*y/(2*m*n) - a**2*x/m
Exercise 50, problem 2
$\displaystyle \left(\frac{a^2b^3}{4y}\right)^3$.
Printed answer:
- $\displaystyle \frac{a^3b^9}{64y^3}$
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem a**3*b**9/(64*y**3)
Exercise 50, problem 20
$\displaystyle \frac{x^2}{yz} \left( \frac{2x}{3} - \frac{3y^2z}{2x} + \frac{x^2y}{m^2z} \right)$.
Printed answer:
- $\displaystyle \frac{2x^3}{3yz}-\frac{3xy}{2}+\frac{x^4}{m^2z^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**3/(3*y*z) - 3*x*y/2 + x**4/(m**2*z**2)
Exercise 50, problem 21
$\displaystyle \left( \frac{a^4}{b^4} + \frac{c^4}{d^4} \right) \left( \frac{a^2}{b^2} + \frac{c^2}{d^2} \right) \left( \frac{ a }{ b } + \frac{ c }{ d } \right) \left( \frac{ a }{ b } - \frac{ c }{ d } \right)$.
Printed answer:
- $\displaystyle \frac{a^8}{b^8}-\frac{c^8}{d^8}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**8/b**8 - c**8/d**8
Exercise 50, problem 22
$\displaystyle \left( \frac{x^2}{16} + \frac{9y^2}{25m^1} \right) \left( \frac{x}{4} + \frac{3y}{5m} \right) \left( \frac{x}{4} - \frac{3y}{5m} \right)$.
Printed answer:
- $\displaystyle \frac{x^4}{256} - \frac{81y^4}{625m^4}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem x**4/256 - 81*y**4/(625*m**4)
Exercise 50, problem 23
$\displaystyle \left( \frac{x}{y} + 1 \right) \left( \frac{x^2}{y^2} - \frac{x}{y} + 1 \right)$.
Printed answer:
- $\displaystyle \frac{x^3}{y^3}+1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3/y**3 + 1
Exercise 50, problem 24
$\displaystyle \left( \frac{a}{b} - 1 \right) \left( \frac{a^2}{b^2} + \frac{a}{b} + 1 \right)$.
Printed answer:
- $\displaystyle \frac{a^3}{b^3}-1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**3/b**3 - 1
Exercise 50, problem 25
$\displaystyle \left( \frac{x}{y} - \frac{3a^2}{2b} \right)$.
Printed answer:
- $\displaystyle \frac{x^2}{y^2}-\frac{3a^2x}{by}+\frac{9a^4}{4b^2}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem x**2/y**2 - 3*a**2*x/(b*y) + 9*a**4/(4*b**2)
Exercise 50, problem 26
$\displaystyle \left( \frac{2y}{c^2} + \frac{9}{2d} \right)^2$.
Printed answer:
- $\displaystyle \frac{4y^2}{c^4}+\frac{18y}{c^2d}+\frac{81}{4d^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*y**2/c**4 + 18*y/(c**2*d) + 81/(4*d**2)
Exercise 50, problem 27
$\displaystyle \left( \frac{a}{b} + 2 \right) \left( \frac{a}{b} - 5 \right)$.
Printed answer:
- $\displaystyle \frac{a^2}{b^2}-\frac{3a}{b}-10$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2/b**2 - 3*a/b - 10
Exercise 50, problem 28
$\displaystyle \left( \frac{x}{y} - \frac{a}{b} \right)^3$.
Printed answer:
- $\displaystyle \frac{x^3}{y^3}-\frac{3ax^2}{by^2}+ \frac{3a^2x}{b^2y}-\frac{a^3}{b^3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3/y**3 - 3*a*x**2/(b*y**2) + 3*a**2*x/(b**2*y) - a**3/b**3
Exercise 50, problem 29
$\displaystyle \frac{x^4}{a^2} - \frac{b^2}{y^4}$
Printed answer:
- $\displaystyle \left(\frac{x^2}{a}-\frac{b}{y^2}\right)\left(\frac{x^2}{a}+ \frac{b}{y^2}\right)$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (x**2/a - b/y**2)*(x**2/a + b/y**2)
Exercise 50, problem 3
$\displaystyle \left(-\frac{2xy}{3a^2m^3}\right)^5)$.
Printed answer:
- $\displaystyle -\frac{32x^5y^5}{243a^{10}m^{15}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -32*x**5*y**5/(243*a**10*m**15)
Exercise 50, problem 30
$\displaystyle \frac{a^2}{b^4} - \frac{x^8}{y^2}$
Printed answer:
- $\displaystyle \left(\frac{a}{b^2}-\frac{x^4}{y}\right)\left(\frac{a}{b^2}+ \frac{x^4}{y}\right)$
verified: the printed answer passed a computed check
How it was checked
- factor: passes (a/b**2 - x**4/y)*(a/b**2 + x**4/y)
Exercise 50, problem 31
$\frac{a^4}{m^4} - \frac{b^4}{x^4}$
Printed answer:
- $\displaystyle \left(\frac{a^2}{m^2}+\frac{b^2}{x^2}\right) \left(\frac{a}{m}+\frac{b}{x}\right) \left(\frac{a}{m}-\frac{b}{x}\right)$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (a**2/m**2 + b**2/x**2)*(a/m + b/x)*(a/m - b/x)
Exercise 50, problem 32
$\displaystyle \frac{8a^3}{y^3} + \frac{b^3}{c^3}$
Printed answer:
- $\displaystyle \left(\frac{2a}{y}+\frac{b}{c}\right) \left(\frac{4a^2}{y^2}-\frac{2ab}{cy}+\frac{b^2}{c^2}\right)$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (2*a/y + b/c)*(4*a**2/y**2 - 2*a*b/(c*y) + b**2/c**2)
Exercise 50, problem 33
$\displaystyle \frac{x^3}{27a^3} - \frac{y^3}{c^3}$
Printed answer:
- $\displaystyle \left(\frac{x}{3a}-\frac{y}{c}\right)\left(\frac{x^2}{9a^2}+ \frac{xy}{3ac}+\frac{y^2}{c^2}\right)$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (x/(3*a) - y/c)*(x**2/(9*a**2) + x*y/(3*a*c) + y**2/c**2)
Exercise 50, problem 34
$\displaystyle \frac{81a^4}{b^4} - \frac{x^4y^4}{625z^4}$
Printed answer:
- $\displaystyle \left(\frac{9a^2}{b^2}+\frac{x^2y^2}{25z^2}\right) \left(\frac{3a}{b}+\frac{xy}{5z}\right) \left(\frac{3a}{b}-\frac{xy}{5z}\right)$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (9*a**2/b**2 + x**2*y**2/(25*z**2))*(3*a/b + x*y/(5*z))*(3*a/b - x*y/(5*z))
Exercise 50, problem 35
$\displaystyle \frac{m^2}{y^2} + \frac{2m}{y} - 15$
Printed answer:
- $\displaystyle \left(\frac{m}{y}-5\right)\left(\frac{m}{y}+3\right)$.
unverified: no computed check settled this one (yet)
How it was checked
- factor: the printed answer does not match the problem (m/y - 5)*(m/y + 3)
Exercise 50, problem 36
$\displaystyle \frac{x^2}{a^2} - \frac{2x}{a} - 8$
Printed answer:
- $\displaystyle \left(\frac{x}{a}-4\right)\left(\frac{x}{a}+2\right)$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (x/a - 4)*(x/a + 2)
Exercise 50, problem 37
$\displaystyle \frac{y^2}{4} + 2 + \frac{4}{y^2}$
Printed answer:
- $\displaystyle \left(\frac{y}{2}+\frac{2}{y}\right)^2$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (y/2 + 2/y)**2
Exercise 50, problem 38
$\displaystyle \frac{x^2}{a^2} + \frac{a^2}{x^2} - 2$
Printed answer:
- $\displaystyle \left(\frac{x}{a}-\frac{a}{x}\right)^2$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (x/a - a/x)**2
Exercise 50, problem 39
A dog can take 2 leaps of $a$ feet each in a second. How many feet can he go in 9 seconds?
Printed answer:
- $18a$ ft.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 18*a
Exercise 50, problem 4
$\displaystyle \left(\frac{x(a-b)}{3a^2b}\right)^4$.
Printed answer:
- $\displaystyle \frac{x^4 \left(a-b \right)^4}{81a^8b^4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4*(a-b)**4/(81*a**8*b**4)
Exercise 50, problem 40
How many weeks would it take to build a stone wall if $\frac{1}{a}$ of it can be built in one day?
Printed answer:
- $\displaystyle \frac{a}{6}$ weeks.
unverified: no computed check settled this one (yet)
How it was checked
- evaluate: the printed answer does not match the problem a/6
Exercise 50, problem 5
$\displaystyle \left(- \frac{5x^2y(a+b^2)^2}{2ab^3(x^2-y)^3}\right)^3$.
Printed answer:
- $\displaystyle -\frac{125x^5y^3\left(a+b^2\right)^6}{8a^3b^9 \left(x^2-y \right)^9}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem -125*x**5*y**3*(a+b**2)**6/(8*a**3*b**9*(x**2-y)**9)
Exercise 50, problem 6
$\displaystyle \left(- \frac{3am^4(2a+3b)^3}{4x^2y^2(m-n)^2}\right)^3$.
Printed answer:
- $\displaystyle -\frac{27a^3m^{12} \left( 2a+3b \right) ^9}{64x^6y^9 \left( m-n \right) ^6}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem -27*a**3*m**12*(2*a+3*b)**9/(64*x**6*y**9*(m-n)**6)
Exercise 50, problem 7
$\displaystyle \left( -\frac{ 3a^5xy^2z^3 }{ 2bc^4d^4 } \right)^4$.
Printed answer:
- $\displaystyle \frac{81a^{20}x^4y^8z^{12}}{16b^4c^{16}d^{16}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 81*a**20*x**4*y**8*z**12/(16*b**4*c**16*d**16)
Exercise 50, problem 8
$\displaystyle \left( -\frac{ 4x^7y^2z^3 }{ 3ab^6d^3 } \right)^4$.
Printed answer:
- $\displaystyle \frac{256x^{28}y^8z^{12}}{81a^4b^{24}d^{12}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 256*x**28*y**8*z**12/(81*a**4*b**24*d**12)
Exercise 50, problem 9
$\displaystyle \sqrt[3]{ \frac{ 8a^3b^6 }{ 27m^6n^9 } }$.
Printed answer:
- $\displaystyle \frac{2ab^2}{3m^2n^3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a*b**2/(3*m**2*n**3)