Fractions

Excerpts

Fractions
The introduction of the same factor into the dividend and divisor does not alter the value of the quotient, and the rejection of the same factor from the dividend and divisor does not alter the value of the quotient.
Fractions
The dividing line between the terms of a fraction has the force of a vinculum affecting the numerator.
Fractions
If, therefore, a *minus sign* precedes the dividing line, as in the preceding Example, and this line is removed, the numerator of the given fraction must be enclosed in a parenthesis preceded by the minus sign, or the sign of every term of the numerator must be changed.
Fractions
By division, $\dfrac{x^{3} - 1}{x + 1} = x^{2} - x + 1 - \dfrac{2}{x + 1}$.
Fractions
The reciprocal of a fraction, therefore, is the fraction inverted.
Fractions
Every mixed expression should first be reduced to a fraction, and every integral expression should be written as a fraction having $1$ for the denominator.

Equations

Fractions
\frac{a}{b} × \frac{c}{d} = \frac{ac}{bd}
The product of two fractions is the product of their numerators over the product of their denominators.
Fractions
\frac{a}{b} × \frac{c}{d} × \frac{e}{f} = \frac{ac}{bd} × \frac{e}{f} = \frac{ace}{bdf}
The product of three fractions is the product of all numerators over the product of all denominators, obtained by applying the two-fraction rule twice.
Fractions
\dfrac{b}{a} × \dfrac{a}{b} = \dfrac{ba}{ab} = 1
The fraction b/a multiplied by its inverse a/b equals 1, so b/a is the reciprocal of a/b.
Fractions
\dfrac{x^{3} - 1}{x - 1} = x^{2} + x + 1
Dividing x cubed minus one by x minus one gives x squared plus x plus one exactly, with no remainder.
Fractions
\dfrac{x^{3} - 1}{x + 1} = x^{2} - x + 1 - \dfrac{2}{x + 1}
Dividing x cubed minus one by x plus one gives an integral part x squared minus x plus one, plus a fraction whose numerator is the remainder 2 and whose denominator is the divisor.

Problems

Exercise 42

Exercise 42, problem 1, p. 90
$\dfrac{2a}{6ab}$.
Printed answer:
- $\dfrac{1}{3b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1/(3*b)
Exercise 42, problem 10, p. 90
$\dfrac{abx - bx^{2}}{acx - cx^{2}}$.
Printed answer:
- $\dfrac{b}{c}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes b/c
Exercise 42, problem 11, p. 90
$\dfrac{4a^{2} - 9b^{2}}{4a^{2} + 6ab}$.
Printed answer:
- $\dfrac{2a - 3b}{2a}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (2*a - 3*b)/(2*a)
Exercise 42, problem 12, p. 90
$\dfrac{3a^{2} + 6a}{a^{2} + 4a + 4}$.
Printed answer:
- $\dfrac{3a}{a + 2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*a/(a + 2)
Exercise 42, problem 13, p. 90
$\dfrac{x^{2} + 5x}{x^{2} + 4x-5}$.
Printed answer:
- $\dfrac{x}{x - 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x/(x - 1)
Exercise 42, problem 14, p. 90
$\dfrac{xy - 3y^{2}}{x^{3} - 27y^{3}}$.
Printed answer:
- $\dfrac{y}{x^{2} + 3xy + 9y^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y/(x**2 + 3*x*y + 9*y**2)
Exercise 42, problem 15, p. 90
$\dfrac{x^{2} + 5x + 4}{x^{2} - x-20}$.
Printed answer:
- $\dfrac{x + 1}{x - 5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x + 1)/(x - 5)
Exercise 42, problem 16, p. 90
$\dfrac{x^{2} + 2x + 1}{x^{2} - x-2}$.
Printed answer:
- $\dfrac{x + 1}{x - 2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x + 1)/(x - 2)
Exercise 42, problem 17, p. 90
$\dfrac{(a + b)^{2} - c^{2}}{a^{2} + ab-ac}$.
Printed answer:
- $\dfrac{a + b + c}{a}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a + b + c)/a
Exercise 42, problem 18, p. 90
$\dfrac{x^{2} + 9x + 20}{x^{2} + 7x + 12}$.
Printed answer:
- $\dfrac{x + 5}{x + 3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x + 5)/(x + 3)
Exercise 42, problem 19, p. 90
$\dfrac{x^{2} - 14x - 15}{x^{2} - 12x - 45}$.
Printed answer:
- $\dfrac{x + 1}{x + 3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x + 1)/(x + 3)
Exercise 42, problem 2, p. 90
$\dfrac{12m^{2}n}{15mn^{2}}$.
Printed answer:
- $\dfrac{4m}{5n}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*m/(5*n)
Exercise 42, problem 3, p. 90
$\dfrac{21m^{2}p^{2}}{28mp^{4}}$.
Printed answer:
- $\dfrac{3m}{4p^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*m/(4*p**2)
Exercise 42, problem 4, p. 90
$\dfrac{3x^{3}y^{2}z}{6xy^{3}z^{2}}$.
Printed answer:
- $\dfrac{x^{2}}{2yz}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2/(2*y*z)
Exercise 42, problem 5, p. 90
$\dfrac{5a^{3}b^{3}c^{3}}{15c^{5}}$.
Printed answer:
- $\dfrac{a^{3}b^{3}}{3c^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**3*b**3/(3*c**2)
Exercise 42, problem 6, p. 90
$\dfrac{34x^{3}y^{4}z^{5}}{51x^{2}y^{3}z^{5}}$.
Printed answer:
- $\dfrac{2xy}{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x*y/3
Exercise 42, problem 7, p. 90
$\dfrac{46m^{2}np^{3}}{69mnp^{4}}$.
Printed answer:
- $\dfrac{2m}{3p}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*m/(3*p)
Exercise 42, problem 8, p. 90
$\dfrac{39a^{2}b^{3}c^{4}}{52a^{5}bc^{3}}$.
Printed answer:
- $\dfrac{3b^{2}c}{4a^{3}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*b**2*c/(4*a**3)
Exercise 42, problem 9, p. 90
$\dfrac{58xy^{4}z^{6}}{87xy^{2}z^{2}}$.
Printed answer:
- $\dfrac{2y^{2}z^{4}}{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*y**2*z**4/3

Exercise 43

Exercise 43, problem 1, p. 91
$\dfrac{a^{2} - b^{2} + 2}{a - b}$.
Printed answer:
- $a + b + \dfrac{2}{a - b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + b + 2/(a - b)
Exercise 43, problem 10, p. 91
$\dfrac{x^{5} - x^{4} + 1}{x^{2} - x - 1}$.
Printed answer:
- $x^{3} + x + 1 + \dfrac{2x + 2}{x^{2} - x - 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 + x + 1 + (2*x + 2)/(x**2 - x - 1)
Exercise 43, problem 2, p. 91
$\dfrac{a^{2} - b^{2} - 2}{a + b}$.
Printed answer:
- $a - b - \dfrac{2}{a + b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a - b - 2/(a + b)
Exercise 43, problem 3, p. 91
$\dfrac{a^{3} - 2a^{2} + 2a + 1}{a^{2} - a - 1}$.
Printed answer:
- $a - 1 + \dfrac{2a}{a^{2} - a - 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a - 1 + 2*a/(a**2 - a - 1)
Exercise 43, problem 4, p. 91
$\dfrac{2x^{2} - 2x + 1}{x + 1}$.
Printed answer:
- $2x - 4 + \dfrac{5}{x + 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x - 4 + 5/(x + 1)
Exercise 43, problem 5, p. 91
$\dfrac{8x^{3}}{2x + 1}$.
Printed answer:
- $4x^{2} - 2x + 1 - \dfrac{1}{2x + 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2 - 2*x + 1 - 1/(2*x + 1)
Exercise 43, problem 6, p. 91
$\dfrac{5x^{3} + 9x^{2} + 3}{x^{2} + x - 1}$.
Printed answer:
- $5x + 4 + \dfrac{x + 7}{x^{2} + x - 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*x + 4 + (x + 7)/(x**2 + x - 1)
Exercise 43, problem 7, p. 91
$\dfrac{a^{3} + a^{2} + 7a - 2}{a^{2} + a + 2}$.
Printed answer:
- $a + \dfrac{5a - 2}{a^{2} + a + 2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + (5*a - 2)/(a**2 + a + 2)
Exercise 43, problem 8, p. 91
$\dfrac{y^{4} + y^{2}x^{2} + x^{4}}{y^{2} + yx + x^{2}}$.
Printed answer:
- $y^{2} - yx + x^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**2 - y*x + x**2
Exercise 43, problem 9, p. 91
$\dfrac{x^{4} - 3x^{3} + x - 1}{x^{2} + x + 1}$.
Printed answer:
- $x^{2} - 4x + 3 + \dfrac{2x - 4}{x^{2} + x + 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 4*x + 3 + (2*x - 4)/(x**2 + x + 1)

Exercise 44

Exercise 44, problem 1, p. 92
$x - y + \dfrac{2xy}{x - y}$.
Printed answer:
- $\dfrac{x^{2} + y^{2}}{x - y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2 + y**2)/(x - y)
Exercise 44, problem 10, p. 92
$a^{2} + 2a - 5 - \dfrac{2a - 1}{3a^{2} + 1}$.
Printed answer:
- $\dfrac{3a^{4} + 6a^{3} - 14a^{2} - 4}{3a^{2} + 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (3*a**4 + 6*a**3 - 14*a**2 - 4)/(3*a**2 + 1)
Exercise 44, problem 2, p. 92
$x + y - \dfrac{2xy}{x + y}$.
Printed answer:
- $\dfrac{x^{2} + y^{2}}{x + y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2 + y**2)/(x + y)
Exercise 44, problem 3, p. 92
$1 - \dfrac{x - y}{x + y}$.
Printed answer:
- $\dfrac{2y}{x + y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*y/(x + y)
Exercise 44, problem 4, p. 92
$a - x - \dfrac{a^{2} + x^{2}}{a - x}$.
Printed answer:
- $-\dfrac{2ax}{a - x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -2*a*x/(a - x)
Exercise 44, problem 5, p. 92
$x + 2 - \dfrac{x^{2} - 4}{x - 3}$.
Printed answer:
- $-\dfrac{x + 2}{x - 3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -(x + 2)/(x - 3)
Exercise 44, problem 6, p. 92
$\dfrac{x - 3}{x - 2} - 2x + 1$.
Printed answer:
- $-\dfrac{2x^{2} - 6x + 5}{x - 2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -(2*x**2 - 6*x + 5)/(x - 2)
Exercise 44, problem 7, p. 92
$\dfrac{x + 3}{x + 2} + x^{2} - x - 1$.
Printed answer:
- $\dfrac{x^{3} + x^{2} - 2x + 1}{x + 2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**3 + x**2 - 2*x + 1)/(x + 2)
Exercise 44, problem 8, p. 92
$2a - 1 + \dfrac{3 - 4a}{a - 3}$.
Printed answer:
- $\dfrac{2a^{2} - 11a + 6}{a - 3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (2*a**2 - 11*a + 6)/(a - 3)
Exercise 44, problem 9, p. 92
$1 - 2a^{2} - \dfrac{a^{2} - a + 2}{a - 1}$.
Printed answer:
- $\dfrac{-2a^{3} + a^{2} + 2a-3}{a - 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (-2*a**3 + a**2 + 2*a - 3)/(a - 1)

Exercise 45

Exercise 45, problem 1a, p. 94
$\dfrac{x}{x - a}$, $\dfrac{x^{2}}{x^{2} - a^{2}}$.
Printed answer:
- $\dfrac{x(x + a)}{(x + a)(x - a)}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes x*(x + a)/((x + a)*(x - a))
Exercise 45, problem 1b, p. 94
$\dfrac{x}{x - a}$, $\dfrac{x^{2}}{x^{2} - a^{2}}$.
Printed answer:
- $\dfrac{x^{2}}{(x + a)(x - a)}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2/((x + a)*(x - a))
Exercise 45, problem 2a, p. 94
$\dfrac{a}{a + b}$, $\dfrac{a^{2}}{a^{2} - b^{2}}$.
Printed answer:
- $\dfrac{a(a - b)}{(a + b)(a - b)}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*(a - b)/((a + b)*(a - b))
Exercise 45, problem 2b, p. 94
$\dfrac{a}{a + b}$, $\dfrac{a^{2}}{a^{2} - b^{2}}$.
Printed answer:
- $\dfrac{a^{2}}{(a + b)(a - b)}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2/((a + b)*(a - b))
Exercise 45, problem 3a, p. 94
$\dfrac{1}{1 + 2a}$, $\dfrac{1}{1 - 4a^{2}}$.
Printed answer:
- $\dfrac{1 - 2a}{(1 + 2a)(1 - 2a)}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes (1 - 2*a)/((1 + 2*a)*(1 - 2*a))
Exercise 45, problem 3b, p. 94
$\dfrac{1}{1 + 2a}$, $\dfrac{1}{1 - 4a^{2}}$.
Printed answer:
- $\dfrac{1}{(1 + 2a)(1 - 2a)}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1/((1 + 2*a)*(1 - 2*a))
Exercise 45, problem 4a, p. 94
$\dfrac{9}{16 - x^{2}}$, $\dfrac{4 - x}{4 + x}$.
Printed answer:
- $\dfrac{9}{(4 + x)(4 - x)}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9/((4 + x)*(4 - x))
Exercise 45, problem 4b, p. 94
$\dfrac{9}{16 - x^{2}}$, $\dfrac{4 - x}{4 + x}$.
Printed answer:
- $\dfrac{(4 - x)^{2}}{(4 + x)(4 - x)}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes (4 - x)**2/((4 + x)*(4 - x))
Exercise 45, problem 5a, p. 94
$\dfrac{a^{2}}{27 - a^{3}}$, $\dfrac{a}{3 - a}$.
Printed answer:
- $\dfrac{a^{2}}{(3 - a)(9 + 3a + a^{2})}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2/((3 - a)*(9 + 3*a + a**2))
Exercise 45, problem 5b, p. 94
$\dfrac{a^{2}}{27 - a^{3}}$, $\dfrac{a}{3 - a}$.
Printed answer:
- $\dfrac{a(9 + 3a + a^{2})}{(3 - a)(9 + 3a + a^{2})}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*(9 + 3*a + a**2)/((3 - a)*(9 + 3*a + a**2))
Exercise 45, problem 6a, p. 94
$\dfrac{1}{x^{2} - 5x + 6}$, $\dfrac{1}{x^{2} - x-6}$.
Printed answer:
- $\dfrac{x + 2}{(x + 2)(x - 2)(x - 3)}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x + 2)/((x + 2)*(x - 2)*(x - 3))
Exercise 45, problem 6b, p. 94
$\dfrac{1}{x^{2} - 5x + 6}$, $\dfrac{1}{x^{2} - x-6}$.
Printed answer:
- $\dfrac{x - 2}{(x + 2)(x - 2)(x - 3)}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x - 2)/((x + 2)*(x - 2)*(x - 3))

Exercise 46

Exercise 46, problem 1, p. 95
$\dfrac{x + 1}{2} + \dfrac{x - 3}{5} + \dfrac{x + 5}{10}$.
Printed answer:
- $\dfrac{4x + 2}{5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (4*x + 2)/5
Exercise 46, problem 2, p. 95
$\dfrac{2x - 1}{3} + \dfrac{x + 5}{4} + \dfrac{x - 4}{6}$.
Printed answer:
- $\dfrac{13x + 3}{12}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (13*x + 3)/12
Exercise 46, problem 3, p. 95
$\dfrac{7x - 1}{6} - \dfrac{3x - 2}{7} + \dfrac{x - 5}{3}$.
Printed answer:
- $\dfrac{5(9x - 13)}{42}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*(9*x - 13)/42
Exercise 46, problem 4, p. 95
$\dfrac{3x - 2}{9} - \dfrac{x - 2}{6} + \dfrac{5x + 3}{4}$.
Printed answer:
- $\dfrac{51x + 31}{36}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (51*x + 31)/36
Exercise 46, problem 5, p. 95
$\dfrac{x - 1}{6} - \dfrac{x - 3}{3} + \dfrac{x - 5}{2}$.
Printed answer:
- $\dfrac{x - 5}{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x - 5)/3
Exercise 46, problem 6, p. 95
$\dfrac{x - 2y}{2x} + \dfrac{x + 5y}{4x} - \dfrac{x + 7y}{8x}$.
Printed answer:
- $\dfrac{5(x - y)}{8x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*(x - y)/(8*x)
Exercise 46, problem 7, p. 95
$\dfrac{5x - 11}{3} - \dfrac{2x - 1}{10} - \dfrac{11x - 5}{15}$.
Printed answer:
- $\dfrac{22x - 97}{30}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (22*x - 97)/30
Exercise 46, problem 8, p. 95
$\dfrac{x - 3}{3x} - \dfrac{x^{2} - 6x}{5x^{2}} - \dfrac{7x^{2} - x^{3}}{15x^{3}}$.
Printed answer:
- $\dfrac{3x - 4}{15x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (3*x - 4)/(15*x)
Exercise 46, problem 9, p. 95
$\dfrac{ac - b^{2}}{ac} - \dfrac{ab - c^{2}}{ab} + \dfrac{a^{2} - bc}{bc}$.
Printed answer:
- $\dfrac{a^{3} - b^{3} + c^{3} - abc}{abc}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a**3 - b**3 + c**3 - a*b*c)/(a*b*c)

Exercise 47

Exercise 47, problem 1, p. 96
$\dfrac{1}{x + 3} + \dfrac{1}{x - 2}$.
Printed answer:
- $\dfrac{2x + 1}{(x + 3)(x - 2)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (2*x + 1)/((x + 3)*(x - 2))
Exercise 47, problem 10, p. 96
$\dfrac{3 - x}{1 - 3x} - \dfrac{3 + x}{1 + 3x} - \dfrac{15x - 1}{1 - 9x^{2}}$.
Printed answer:
- $\dfrac{1 + x}{1 - 9x^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (1 + x)/(1 - 9*x**2)
Exercise 47, problem 11, p. 96
$\dfrac{1}{a} - \dfrac{1}{a + 3} + \dfrac{3}{a + 1}$.
Printed answer:
- $\dfrac{3(a^{2} + 4a + 1)}{a(a + 1)(a + 3)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*(a**2 + 4*a + 1)/(a*(a + 1)*(a + 3))
Exercise 47, problem 12, p. 96
$\dfrac{x}{x - 1} - \dfrac{1} - \dfrac{1}{x + 1}$.
Printed answer:
- $\dfrac{2}{x^{2} - 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2/(x**2 - 1)
Exercise 47, problem 13, p. 96
$\dfrac{x + 1}{x + 2} + \dfrac{x - 2}{x - 3} + \dfrac{2x + 7}{x^{2} - x - 6}$.
Printed answer:
- $\dfrac{2x^{2}}{(x + 2)(x - 3)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**2/((x + 2)*(x - 3))
Exercise 47, problem 14, p. 96
$\dfrac{1}{x(x - 1)} - \dfrac{2}{x^{2} - 1} + \dfrac{1}{x(x + 1)}$.
Printed answer:
- $0$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 0
Exercise 47, problem 2, p. 96
$\dfrac{1}{x + 1} + \dfrac{1}{x - 1}$.
Printed answer:
- $\dfrac{2x}{x^{2} - 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x/(x**2 - 1)
Exercise 47, problem 3, p. 96
$\dfrac{4}{x - 8} - \dfrac{1}{x + 2}$.
Printed answer:
- $\dfrac{3x + 16}{(x - 8)(x + 2)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (3*x + 16)/((x - 8)*(x + 2))
Exercise 47, problem 4, p. 96
$\dfrac{a + x}{a - x} - \dfrac{a - x}{a + x}$.
Printed answer:
- $\dfrac{4ax}{a^{3} - x^{2}}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 4*a*x/(a**3 - x**2)
Exercise 47, problem 5, p. 96
$\dfrac{x}{x - a} - \dfrac{x^{2}}{x^{2} - a^{2}}$.
Printed answer:
- $\dfrac{ax}{x^{2} - a^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*x/(x**2 - a**2)
Exercise 47, problem 6, p. 96
$\dfrac{4a^{2} + b^{2}}{4a^{2} - b^{2}} - \dfrac{2a + b}{2a - b}$.
Printed answer:
- $-\dfrac{4ab}{4a^{2} - b^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -4*a*b/(4*a**2 - b**2)
Exercise 47, problem 7, p. 96
$\dfrac{7}{9 - a^{2}} - \dfrac{1}{3 + a} - \dfrac{1}{3 - a}$.
Printed answer:
- $\dfrac{1}{9 - a^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1/(9 - a**2)
Exercise 47, problem 8, p. 96
$\dfrac{1}{a - b} - \dfrac{1}{a + b} - \dfrac{b}{a^{2} - b^{2}}$.
Printed answer:
- $\dfrac{b}{a^{2} - b^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes b/(a**2 - b**2)
Exercise 47, problem 9, p. 96
$\dfrac{2}{x - 2} - \dfrac{2}{x + 2} + \dfrac{5x}{x^{2} - 4}$.
Printed answer:
- $\dfrac{5x + 8}{x^{2} - 4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (5*x + 8)/(x**2 - 4)

Exercise 48

Exercise 48, problem 1, p. 97
$\dfrac{1}{2x + 1} + \dfrac{1}{2x - 1} - \dfrac{4x}{4x^{2} - 1}$.
Printed answer:
- $0$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 0
Exercise 48, problem 2, p. 97
$\dfrac{a^{2} + b^{2}}{a^{2} - b^{2}} + \dfrac{a}{a + b} - \dfrac{b}{a - b}$.
Printed answer:
- $\dfrac{2a}{a + b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a/(a + b)
Exercise 48, problem 3, p. 97
$\dfrac{3a}{1 - a^{2}} + \dfrac{2}{1 - a} - \dfrac{2}{1 + a}$.
Printed answer:
- $\dfrac{7a}{1 - a^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 7*a/(1 - a**2)
Exercise 48, problem 4, p. 97
$\dfrac{1}{2x + 5y} - \dfrac{3x}{4x^{2} - 25y^{2}} + \dfrac{1}{2x + 5y}$.
Printed answer:
- $\dfrac{x - 10y}{4x^{2} - 25y^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x - 10*y)/(4*x**2 - 25*y**2)
Exercise 48, problem 5, p. 97
$\dfrac{1}{x + 4y} - \dfrac{8y}{x^{2} - 16y^{2}} + \dfrac{1}{x - 4y}$.
Printed answer:
- $\dfrac{2}{x + 4y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2/(x + 4*y)
Exercise 48, problem 6, p. 97
$\dfrac{3}{2x - 3} - \dfrac{2}{2x + 3} - \dfrac{3}{4x^{2} - 9}$.
Printed answer:
- $\dfrac{2(x + 6)}{4x^{2} - 9}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*(x + 6)/(4*x**2 - 9)

Exercise 49

Exercise 49, problem 1, p. 100
$\dfrac{15a^{2}}{7b^{2}} × \dfrac{28ab}{9a^{3}c}$.
Printed answer:
- $\dfrac{20}{3bc}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 20/(3*b*c)
Exercise 49, problem 10, p. 100
$\dfrac{a^{2} - 100}{a^{2} - 9} × \dfrac{a - 3}{a - 10}$.
Printed answer:
- $\dfrac{a + 10}{a + 3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a + 10)/(a + 3)
Exercise 49, problem 11, p. 100
$\dfrac{9x^{2} - 4y^{2}}{x^{2} - 4} × \dfrac{x + 2}{3x - 2y}$.
Printed answer:
- $\dfrac{3x + 2y}{x - 2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (3*x + 2*y)/(x - 2)
Exercise 49, problem 12, p. 100
$\dfrac{25a^{2} - b^{2}}{16a^{2} - 9b^{2}} ÷ \dfrac{5a - b}{4a - 3b}$.
Printed answer:
- $\dfrac{5a + b}{4a + 3b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (5*a + b)/(4*a + 3*b)
Exercise 49, problem 13, p. 100
$\dfrac{x^{2} - 49}{(a + b)^{2} - c^{2}} ÷ \dfrac{x + 7}{(a + b) - c}$.
Printed answer:
- $\dfrac{x - 7}{a + b + c}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x - 7)/(a + b + c)
Exercise 49, problem 14, p. 100
$\dfrac{x^{2} + 2x + 1}{x^{2} - 25} ÷ \dfrac{x + 1}{x^{2} + 5x}$.
Printed answer:
- $\dfrac{x(x + 1)}{x - 5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x*(x + 1)/(x - 5)
Exercise 49, problem 15, p. 100
$\dfrac{a^{2} + 3a + 2}{a^{2} + 5a + 6} × \dfrac{a^{2} + 7a + 12}{a^{2} + 9a + 20}$.
Printed answer:
- $\dfrac{a + 1}{a + 5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a + 1)/(a + 5)
Exercise 49, problem 16, p. 100
$\dfrac{y^{2} - y-30}{y^{2} - 36} × \dfrac{y^{2} - y-2}{y^{2} + 3y-10} × \dfrac{y^{2} + 6y}{y^{2} + y}$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1
Exercise 49, problem 17, p. 100
$\dfrac{x^{2} - 2x + 1}{x^{2} - y^{2}} × \dfrac{x^{2} + 2xy + y^{2}}{x - 1} ÷ \dfrac{x^{2} - 1}{x^{2} - xy}$.
Printed answer:
- $\dfrac{x(x + y)}{x + 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x*(x + y)/(x + 1)
Exercise 49, problem 18, p. 100
$\dfrac{a^{2} - b^{2}}{a^{2} - 3ab + 2b^{2}} × \dfrac{ab - 2b^{2}}{a^{2} + ab} ÷ \dfrac{(a - b)^{2}}{a(a - b)}$.
Printed answer:
- $\dfrac{b}{a - b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes b/(a - b)
Exercise 49, problem 19, p. 100
$\dfrac{(a + b)^{2} - c^{2}}{a^{2} + ab-ac} × \dfrac{a^{2}b^{2}c^{2}}{a^{2} + ab + ac} ÷ \dfrac{b^{2}c^{2}}{abc}$.
Printed answer:
- $abc$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*b*c
Exercise 49, problem 2, p. 100
$\dfrac{3x^{2}y^{2}z^{3}}{4a^{2}b^{2}c^{2}} × \dfrac{8a^{3}b^{2}c^{2}}{9x^{2}yz^{3}}$.
Printed answer:
- $\dfrac{2ay}{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a*y/3
Exercise 49, problem 20, p. 100
$\dfrac{x^{2} + 7xy + 10y^{2}}{x^{2} + 6xy + 5y^{2}} × \dfrac{x + 1}{x^{2} + 4x + 4} ÷ \dfrac{1}{x + 2}$.
Printed answer:
- $\dfrac{(x + 2y)(x + 1)}{(x + y)(x + 2)}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x + 2*y)*(x + 1)/((x + y)*(x + 2))
Exercise 49, problem 3, p. 100
$\dfrac{5m^{2}n^{2}p^{4}}{3x^{2}yz^{3}} × \dfrac{21xyz^{2}}{20m^{2}n^{2}p^{2}}$.
Printed answer:
- $\dfrac{7p^{2}}{4xz}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 7*p**2/(4*x*z)
Exercise 49, problem 4, p. 100
$\dfrac{16a^{4}b^{2}c^{3}}{21m^{2}x^{3}y^{4}} × \dfrac{3m^{3}x^{3}y^{4}}{8a^{2}b^{2}c^{2}}$.
Printed answer:
- $\dfrac{2a^{2}cm}{7}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a**2*c*m/7
Exercise 49, problem 5, p. 100
$\dfrac{2a}{bc} × \dfrac{3b}{ac} × \dfrac{5c}{ab}$.
Printed answer:
- $\dfrac{30}{abc}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 30/(a*b*c)
Exercise 49, problem 6, p. 100
$\dfrac{2a^{3}}{3bc} × \dfrac{3b^{3}}{5ac} × \dfrac{5c^{3}}{2ab}$.
Printed answer:
- $abc$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*b*c
Exercise 49, problem 7, p. 100
$\dfrac{5abc^{3}}{3x^{2}} ÷ \dfrac{10ac^{3}}{6bx^{2}}$.
Printed answer:
- $b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes b**2
Exercise 49, problem 8, p. 100
$\dfrac{x^{2} - a^{2}}{x^{2} - 4a^{2}} × \dfrac{x + 2a}{x - a}$.
Printed answer:
- $\dfrac{x + a}{x - 2a}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x + a)/(x - 2*a)
Exercise 49, problem 9, p. 100
$\dfrac{x^{2}y^{2} + 3xy}{4c^{2} - 1} × \dfrac{2c + 1}{xy + 3}$.
Printed answer:
- $\dfrac{xy}{2c - 1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x*y/(2*c - 1)

Exercise 50

Exercise 50, problem 1, p. 102
$\dfrac{\dfrac{x}{b} + \dfrac{y}{b}}{\dfrac{z}{b}}$.
Printed answer:
- $\dfrac{x + y}{z}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x + y)/z
Exercise 50, problem 10, p. 102
$\dfrac{x + 3 + \dfrac{2}{x}}{1 + \dfrac{3}{x} + \dfrac{2}{x^{2}}}$.
Printed answer:
- $x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x
Exercise 50, problem 11, p. 102
$\dfrac{\dfrac{1}{x} - \dfrac{2}{x^{2}} + \dfrac{1}{x^{3}}}{\dfrac{(1 - x)^{2}}{x^{2}}}$.
Printed answer:
- $\dfrac{1}{x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1/x
Exercise 50, problem 12, p. 102
$\dfrac{x^{2} - x-6}{1 - \dfrac{4}{x^{2}}}$.
Printed answer:
- $\dfrac{x^{2}(x - 3)}{x - 2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2*(x - 3)/(x - 2)
Exercise 50, problem 13, p. 102
$\dfrac{a^{2} - a + \dfrac{a - 1}{a + 1}}{a + \dfrac{1}{a + 1}}$.
Printed answer:
- $a - 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a - 1
Exercise 50, problem 14, p. 102
$\dfrac{\dfrac{4a(a - x)}{a^{2} - x^{2}}}{\dfrac{a - x}{a + x}}$.
Printed answer:
- $\dfrac{4a}{a - x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a/(a - x)
Exercise 50, problem 2, p. 102
$\dfrac{x + \dfrac{y}{4}}{x - \dfrac{y}{3}}$.
Printed answer:
- $\dfrac{12x + 3y}{12x - 4y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (12*x + 3*y)/(12*x - 4*y)
Exercise 50, problem 3, p. 102
$\dfrac{\dfrac{ab}{7} - 3d}{3c - \dfrac{ab}{d}}$.
Printed answer:
- $\dfrac{abd - 21d^{2}}{21cd - 7ab}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a*b*d - 21*d**2)/(21*c*d - 7*a*b)
Exercise 50, problem 4, p. 102
$\dfrac{1 + \dfrac{1}{x + 1}}{1 - \dfrac{1}{x - 1}}$.
Printed answer:
- $\dfrac{x^{2} + x - 2}{x^{2} - x - 2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2 + x - 2)/(x**2 - x - 2)
Exercise 50, problem 5, p. 102
$\dfrac{\dfrac{2m + x}{m + x} - 1}{1 - \dfrac{x}{m + x}}$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1
Exercise 50, problem 6, p. 102
$\dfrac{\dfrac{x + y}{x^{2} - y^{2}}}{\dfrac{x - y}{x + y}}$.
Printed answer:
- $\dfrac{x + y}{x^{2} - 2xy + y^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x + y)/(x**2 - 2*x*y + y**2)
Exercise 50, problem 7, p. 102
$\dfrac{a + \dfrac{ab}{a - b}}{a - \dfrac{ab}{a + b}}$.
Printed answer:
- $\dfrac{a + b}{a - b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a + b)/(a - b)
Exercise 50, problem 8, p. 102
$\dfrac{9a^{2} - 64}{a - 1 - \dfrac{a + 4}{4}}$.
Printed answer:
- $4(3a + 8)$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*(3*a + 8)
Exercise 50, problem 9, p. 102
$\dfrac{\dfrac{1}{x} + \dfrac{1}{y}}{\dfrac{1}{x} - \dfrac{1}{y}}$.
Printed answer:
- $\dfrac{y + x}{y - x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (y + x)/(y - x)