DIVISION

Excerpts

DIVISION
**Division** is the process by which, when a product is given and one factor known, the other factor is found.
DIVISION
What is the relation of the dividend to the divisor and quotient? What factors must the dividend contain? What factors must the quotient contain?
DIVISION
Why stop the work at the point given? What is the complete quotient? Is the dividend exactly divisible by the divisor? When is one number exactly divisible by another?
DIVISION
To divide a polynomial by a monomial, divide each term of the dividend by the divisor, and add the quotients.
DIVISION
To divide a polynomial by a polynomial, arrange the terms of the dividend, and divisor similarly, divide the first term of the dividend by the first term of the divisor for the first term of the quotient, multiply the divisor by the quotient and subtract the product from the dividend; divide the first term of the remainder by the first term of the divisor for the next term of the quotient, multiply and subtract as before; continue this work of dividing, multiplying, and subtracting until there is no remainder or until the first term of the remainder is not divisible by the first term of the divisor.
DIVISION
Since division is the reverse of multiplication, let us consider an example in the multiplication of polynomials.
DIVISION
is indicated by the radical sign and index. When no index is expressed, *two* is understood.

Equations

DIVISION
(a + b)^{2} = a^{2} + 2ab + b^{2} = a^{2} + (2a + b)b
The square of a sum of two quantities equals the square of the first, plus twice their product, plus the square of the second.
DIVISION
(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} = a^{3} + (3 a^{2} + 3 a b + b^{2}) b
The cube of a sum of two quantities expands to the cube of the first, plus three times the square of the first times the second, plus three times the first times the square of the second, plus the cube of the second.

Problems

Exercise 24

Exercise 24, problem 1
$15x^2 y$ by $3x$.
Printed answer:
- $5xy$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*x*y
Exercise 24, problem 10
$\frac{1}{5} x^3 y^4$ by $-\frac{1}{15} xy^3$.
Printed answer:
- $-3x^2y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -3*x**2*y
Exercise 24, problem 11
$-45x^5 y^7 z$ by $9x^2 y^4 z$.
Printed answer:
- $-5x^3 y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -5*x**3*y**3
Exercise 24, problem 12
$60a^4 bc^{11}$ by $-4ab^2 c^7$.
Printed answer:
- $-5a^2 c^4$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem -5*a**2*c**4
Exercise 24, problem 13
$-\frac{2}{3} x^7 y^2$ by $-\frac{5}{6} x^4 y$.
Printed answer:
- $\frac{4}{5} x^3y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(4,5)*x**3*y
Exercise 24, problem 14
$\frac{3}{4} a^5 m^4 n^3$ by $-\frac{1}{4} a^2 mn^3$.
Printed answer:
- $-3a^3 m^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -3*a**3*m**3
Exercise 24, problem 15
$5m^4 n^2 x^5$ by $\frac{5}{8} mn^2 x$.
Printed answer:
- $8m^3 x^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 8*m**3*x**4
Exercise 24, problem 16
$4x^3 y^2 z^8$ by $-\frac{2}{3} xz^5$.
Printed answer:
- $-6x^2 y^2 z^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -6*x**2*y**2*z**3
Exercise 24, problem 17
$10(x + y)^4 z^3$ by $5(x + y)^2 z$.
Printed answer:
- $2(x + y)^2 z^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*(x + y)**2*z**2
Exercise 24, problem 18
$15(a - b)^3 x^2$ by $3(a - b)x$.
Printed answer:
- $5(a - b)^2 x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*(a - b)**2*x
Exercise 24, problem 19
Simplify $(-x^{2}y^{3}z^{2}) \times (-x^{4}y^{5}z^{4}) \div 2x^{3}y^{3}z^{4}$.
Printed answer:
- $\frac{1}{2} x^3 y^3 z^2$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem Rational(1,2)*x**3*y**3*z**2
Exercise 24, problem 2
$39a b^2$ by $3b$.
Printed answer:
- $13ab$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 13*a*b
Exercise 24, problem 20
Simplify $a^{5}b^{2}c \times (-a^{3}b^{3}c^{3}) \div 3(a^{3}bc^{2})^{2}$.
Printed answer:
- $-\frac{1}{3} a^2 b^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -Rational(1,3)*a**2*b**3
Exercise 24, problem 21
Expand $(x^{3}y^{2}-3xy)^{3}$.
Printed answer:
- $x^9 y^6 - 9x^7 y^5 + 27x^5 y^4 - 27 x^3 y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**9*y**6 - 9*x**7*y**5 + 27*x**5*y**4 - 27*x**3*y**3
Exercise 24, problem 22
If a man can ride one mile for $a$ cents, how far can two men ride for $b$ cents?
Printed answer:
- $\frac{b}{2a}$ miles.
verified: the printed answer passed a computed check
How it was checked
- solve: passes b/(2*a)
Exercise 24, problem 23
In how many days can $x$ men earn as much as 8 men in $y$ days?
Printed answer:
- $\frac{8y}{x}$ days.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 8*y/x
Exercise 24, problem 24
$a$ times $b$ is how many times $c$?
Printed answer:
- $\frac{ab}{c}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes a*b/c
Exercise 24, problem 3
$27a^3 b^3 c$ by $9ab^2 c$.
Printed answer:
- $3a^2$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 3*a**2
Exercise 24, problem 4
$35x^4 y^4 z$ by $7x^2 yz$.
Printed answer:
- $5x^2 y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*x**2*y**3
Exercise 24, problem 5
$-51cx^3 y$ by $3cyx^2$.
Printed answer:
- $-17x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -17*x
Exercise 24, problem 6
$121x^3 y^3 z$ by $-11y^2 z$.
Printed answer:
- $-11x^3 y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -11*x**3*y
Exercise 24, problem 7
$-28x^2 y^2 z^2$ by $-7x y^2$.
Printed answer:
- $4xz^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x*z**2
Exercise 24, problem 8
$-36a^3 b^2 c^4$ by $-4a b^2 c^2$.
Printed answer:
- $9a^2c^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*a**2*c**2
Exercise 24, problem 9
$\frac{1}{3} a^4 b^5$ by $\frac{1}{6} a^2 b^2$.
Printed answer:
- $2a^2 b^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a**2*b**3

Exercise 25

Exercise 25, problem 1
$18a^{4}b^{3}-42a^{3}b^{2}+90a^{6}bx$ by $6a^{3}b$.
Printed answer:
- $3ab^2 - 7b + 15a^3 x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*a*b**2 - 7*b + 15*a**3*x
Exercise 25, problem 10
$24y^{3}+32y^{7}-48y^{6}$ by $3y^{3}$.
Printed answer:
- $8 + \frac{32}{3} y^4 - 16y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 8 + Rational(32,3)*y**4 - 16*y**3
Exercise 25, problem 11
$\frac{1}{4}a^2x-\frac{1}{16}abx - \frac{3}{8}acx$ by $\frac{3}{8}ax$.
Printed answer:
- $\frac{2}{3} a - \frac{1}{6} b - c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(2,3)*a - Rational(1,6)*b - c
Exercise 25, problem 12
$-\frac{5}{2}x^2+\frac{5}{3}xy+\frac{10}{3}x$ by $-\frac{5}{6}x$.
Printed answer:
- $3x - 2y - 4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x - 2*y - 4
Exercise 25, problem 13
An army was drawn up with $x$ men in front and $y$ men deep. How many men were there in the army?
Printed answer:
- $xy$ men.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x*y
Exercise 25, problem 14
In how many minutes will a train go $x$ miles at the rate of $a$ miles an hour?
Printed answer:
- $\frac{60x}{a}$ minutes.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 60*x/a
Exercise 25, problem 15
How many apples at $x$ cents apiece can be bought for $b$ dollars?
Printed answer:
- $\frac{100b}{x}$ apples.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 100*b/x
Exercise 25, problem 2
$10x^{5}y^{2}+6x^{3}y^{2}-18x^{4}y^{4}$ by $2x^{3}y$.
Printed answer:
- $5x^2 y + 3y - 9xy^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*x**2*y + 3*y - 9*x*y**3
Exercise 25, problem 3
$72x^{5}y^{6}-36x^{4}y^{3}-18x^{2}y^{2}$ by $9x^{2}y$.
Printed answer:
- $8x^3 y^5 - 4x^2 y^2 - 2y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 8*x**3*y**5 - 4*x**2*y**2 - 2*y
Exercise 25, problem 4
$169a^{4}b-117a^{3}b^{2}+91a^{2}b$ by $13a^{2}$.
Printed answer:
- $13a^2 b - 9ab^2 + 7b$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 13*a**2*b - 9*a*b**2 + 7*b
Exercise 25, problem 5
$-2a^{5}x^{3}+\frac{7}{2}a^{4}x^{4}$ by $\frac{7}{3}a^{3}x$.
Printed answer:
- $-\frac{6}{7} a^2x^2 + \frac{3}{2} ax^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -Rational(6,7)*a**2*x**2 + Rational(3,2)*a*x**3
Exercise 25, problem 6
$\frac{1}{2}x^{5}y^{2}-3x^{3}y^{4}$ by $-\frac{3}{2}x^3y^{2}$.
Printed answer:
- $-\frac{1}{3} x^2 + 2y^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -Rational(1,3)*x**2 + 2*y**2
Exercise 25, problem 7
$32x^{3}y^{4}z^{6}-24x^{5}y^{4}z^{4}+8x^{4}y^{2}$ by $-8x^{3}y$.
Printed answer:
- $-4y^3 z^3 + 3x^2 y^3 z^4 - xy$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem -4*y**3*z**3 + 3*x**2*y**3*z**4 - x*y
Exercise 25, problem 8
$120a^{4}b^{3}c^{5}-186a^{5}b^{7}c^{6}$ by $6a^{3}b^{3}c^{4}$.
Printed answer:
- $20ac - 31a^2 b^4 c^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 20*a*c - 31*a**2*b**4*c**2
Exercise 25, problem 9
$4x^{8}-10x^{5}+2x^{6}-16x^{3}-6x^{4}$ by $4x^{2}$.
Printed answer:
- $x^6 - \frac{5}{2} x^3 + \frac{1}{2} x^4 - 4x - \frac {3}{2} x^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**6 - Rational(5,2)*x**3 + Rational(1,2)*x**4 - 4*x - Rational(3,2)*x**2

Exercise 26

Exercise 26, problem 1
$x^2+8x-105$ by $x+15$.
Printed answer:
- $x - 7$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x - 7
Exercise 26, problem 10
$81x^8-y^4$ by $3x^2-y$.
Printed answer:
- $27x^5 + 9x^4y + 3x^2y^2 + y^3$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 27*x**5 + 9*x**4*y + 3*x**2*y**2 + y**3
Exercise 26, problem 11
$x^5-x^4y-2x^3y^2-5x^2y^3-17xy^4-12y^5$ by $x^2-2xy-3y^2$.
Printed answer:
- $x^3 + x^2y + 3xy^2 + 4y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 + x**2*y + 3*x*y**2 + 4*y**3
Exercise 26, problem 12
$a^5+a^4b-14a^3b^2+15a^2b^3-4b^3$ by $a^2-3ab+2b^2$.
Printed answer:
- $a^3 + 4a^2b - 3ab^2 - 2b^3$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem a**3 + 4*a**2*b - 3*a*b**2 - 2*b**3
Exercise 26, problem 13
$x^6 - 5x^3 + 3 + 5x^4 - 10x - x^5 + 10x^2$ by $x^2 + 3 - x$.
Printed answer:
- $x^4 + 2x^2 - 3x + 1.$
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4 + 2*x**2 - 3*x + 1
Exercise 26, problem 14
$x^6 - 2x^3 - 2 + x - 3x^5 + 2x^4 - 5x^2$ by $x^3 + 2 + x$.
Printed answer:
- $x^3 - 3x^2 + x - 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 - 3*x**2 + x - 1
Exercise 26, problem 15
$a^5 - a - 2a^2 - a^3$ by $a + a^3 + a^2$.
Printed answer:
- $a^2 - a - 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 - a - 1
Exercise 26, problem 16
$x^6 - 2x^3 - x^2 - x^4$ by $1 + xx^2 + x$.
Printed answer:
- $x^4 - x^3 - x^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4 - x**3 - x**2
Exercise 26, problem 17
$a^{11} - a^2$ by $a^3 - 1$.
Printed answer:
- $a^8 + a^5 + a^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**8 + a**5 + a**2
Exercise 26, problem 18
$a^{12} - a^4$ by $a^2 + 1$.
Printed answer:
- $a^10 - a^8 + a^6 - a^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**10 - a**8 + a**6 - a**4
Exercise 26, problem 19
$x^4 + 4y^4$ by $x^2 - 2xy + y^2$.
Printed answer:
- $x^2 + 2xy + 2y^2$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem x**2 + 2*x*y + 2*y**2
Exercise 26, problem 2
$x^2+8x-33$ by $x+11$.
Printed answer:
- $x - 3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x - 3
Exercise 26, problem 20
$4a^4 + 81b^4$ by $2a^2 + 6ab + 9b^2$.
Printed answer:
- $2a^2 - 6ab + 9b^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a**2 - 6*a*b + 9*b**2
Exercise 26, problem 21
$\frac{1}{2}x^4 + \frac{3}{4}x^3y - \frac{1}{3}x^2y^2 + \frac{7}{6}xy^3 - \frac{1}{3}y^4$ by $x^2 - \frac{1}{2}xy + y^2$.
Printed answer:
- $\frac{1}{2} x^2 + xy - \frac{1}{3} y^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(1,2)*x**2 + x*y - Rational(1,3)*y**2
Exercise 26, problem 22
$\frac{2}{9}y^3 - \frac{5}{36}x^2y + \frac{1}{6}xy^2 + \frac{1}{6}x^3$ by $\frac{1}{2}x + \frac{1}{3}y$.
Printed answer:
- $\frac{1}{3} x^2 - \frac{1}{2} xy + \frac {2}{3} y^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(1,3)*x**2 - Rational(1,2)*x*y + Rational(2,3)*y**2
Exercise 26, problem 23
$\frac{1}{4}(x - y)^5 - (x - y)^3 - \frac{1}{2}(x - y)^2 - \frac{1}{16}(x - y)$ by $\frac{1}{2}(x - y)^2 + (x - y) +\frac{1}{4}$.
Printed answer:
- $\frac{1}{2}\left(x - y\right)^3 - \left(x - y\right)^2 - \frac{1}{4}(x - y)$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(1,2)*(x - y)**3 - (x - y)**2 - Rational(1,4)*(x - y)
Exercise 26, problem 24
$16x^8 - 81y^4$ by $27y^3 + 18x^2y^2 + 8x^6 + 12x^4y$.
Printed answer:
- $2x^2 - 3y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**2 - 3*y
Exercise 26, problem 25
$4x^4 - 10x^2 + 6$ by $x + 1$.
Printed answer:
- $4x^3 - 4x^2 - 6x + 6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**3 - 4*x**2 - 6*x + 6
Exercise 26, problem 26
$4a^4 - 5a^2b^2 + b^4$ by $2a - b$.
Printed answer:
- $2a^3 + a^2b - 2ab^2 - b^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a**3 + a**2*b - 2*a*b**2 - b**3
Exercise 26, problem 27
$a^6 - b^6 + a^4 + b^4 + a^2b^2$ by $a^2 - b^2 + 1$.
Printed answer:
- $a^4 + a^2b^2 + b^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4 + a**2*b**2 + b**4
Exercise 26, problem 28
$x^6 - y^6 - x^4 - y^4 - x^2y^2$ by $x^2 - y^2 - 1$.
Printed answer:
- $x^4 + x^2y^2 + y^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4 + x**2*y**2 + y**4
Exercise 26, problem 29
$81a^{12} - 16b^8$ by $12a^3b^4 - 8b^6 - 18a^6b^2 + 27a^9$.
Printed answer:
- $3a^3 + 2b^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*a**3 + 2*b**2
Exercise 26, problem 3
$x^4+x^2-20$ by $x^2-4$.
Printed answer:
- $x^2 + 5$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 5
Exercise 26, problem 30
$1 + 3x$ by $1 + x$ to four terms of quotient.
Printed answer:
- $1 + 2x - 2x^2 + 2x^3 - \text{etc}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 26, problem 31
$1 - 2a$ by $1 - a$ to four terms of quotient.
Printed answer:
- $1 - a - a^2 - a^3 - \text{etc}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 26, problem 32
$4 + a$ by $2 - a$ to four terms.
Printed answer:
- $2 + \frac{3}{2} a + \frac{3}{4} a^2 + \frac{3}{8} a^3 + \text{etc}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 26, problem 33
$9 - x$ by $3 + x$ to four terms.
Printed answer:
- $3 - \frac{4}{3} x + \frac{4}{9} x^2 - \frac{4}{27} x^3 + \text{etc}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 26, problem 34
If a boy can do a piece of work in $x$ minutes, how many hours would it take him to perform $12$ times as much work?
Printed answer:
- $\displaystyle \frac{x}{5}$ hrs.
verified: the printed answer passed a computed check
How it was checked
- identity: passes, with the problem read into an equation x/5
Exercise 26, problem 35
A man has $x$ dollars, $y$ acres of land worth $m$ dollars an acre, and $c$ houses each worth $b$ dollars. What is my share if I am one of $n$ heirs?
Printed answer:
- $\displaystyle \frac{x + my + bc}{n}$ dols.
verified: the printed answer passed a computed check
How it was checked
- identity: passes, with the problem read into an equation (x + m*y + b*c)/n
Exercise 26, problem 36
A storekeeper mixed $m$ pounds of coffee worth $a$ cents a pound with $p$ pounds worth $b$ cents a pound. How much is the mixture worth per pound?
Printed answer:
- $\displaystyle \frac{am + bp}{m + p}$ cts.
verified: the printed answer passed a computed check
How it was checked
- identity: passes, with the problem read into an equation (a*m + b*p)/(m + p)
Exercise 26, problem 37
If John is $y$ years old, how old was he 11 years ago?
Printed answer:
- $y - 11$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 26, problem 4
$y^4-y^2-30$ by $y^2+5$.
Printed answer:
- $y^2 - 6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**2 - 6
Exercise 26, problem 5
$x^4-31x^2+9$ by $x^2+5x-3$.
Printed answer:
- $x^2 - 5x - 3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 5*x - 3
Exercise 26, problem 6
$a^4-12a^2+16$ by $a^2-2a-4$.
Printed answer:
- $a^2 + 2a - 4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 + 2*a - 4
Exercise 26, problem 7
$x^3-y^3$ by $x-y$.
Printed answer:
- $x^2 + xy + y^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + x*y + y**2
Exercise 26, problem 8
$a^3+b^3$ by $a+b$.
Printed answer:
- $a^2 - ab + b^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 - a*b + b**2
Exercise 26, problem 9
$16a^4-81b^4$ by $2a-3b$.
Printed answer:
- $8a^3 + 12a^2 b + 18ab^2 + 27b^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 8*a**3 + 12*a**2*b + 18*a*b**2 + 27*b**3

Exercise 27

Exercise 27, problem 1
$\sqrt{16 a^{2} b^{6}}$.
Printed answer:
- $4ab^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a*b**3
Exercise 27, problem 10
$\sqrt[3]{\frac{8}{27} a^{9} b^{6}}$.
Printed answer:
- $\frac(2)(3) a^3b^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(2,3)*a**3*b**2
Exercise 27, problem 11
$\sqrt[3]{-\frac{27}{64} x^{9} y^{12}}$.
Printed answer:
- $- \frac{3}{4} x^3y^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -Rational(3,4)*x**3*y**4
Exercise 27, problem 12
$\sqrt[5]{-\frac{32}{243} a^{5} b^{15}}$.
Printed answer:
- $- \frac{2}{3} ab^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -Rational(2,3)*a*b**3
Exercise 27, problem 13
$\sqrt[6]{x^{12} \left(a - b \right)^{6}}$.
Printed answer:
- $x^2(a - b)$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2*(a - b)
Exercise 27, problem 14
$\sqrt[3]{a^9 \left(x^{2} + y^{2} \right)^{3}}$.
Printed answer:
- $a^3(x^2 + y^2)$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**3*(x**2 + y**2)
Exercise 27, problem 15
$\sqrt{4 a^{2} b^{6} \left(x^{2} - y \right)^{4}}$.
Printed answer:
- $2ab^3\left(x^2 - y\right)^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a*b**3*(x**2 - y)**2
Exercise 27, problem 16
$\sqrt{16 x^{4} y^{2} \left(m^{3} + y \right)^{6}}$.
Printed answer:
- $4x^2y\left(m^3 + y\right)^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2*y*(m**3 + y)**3
Exercise 27, problem 17
$\sqrt{\frac{1}{9} a^{4} b^{6}} - \sqrt[3]{\frac{1}{8} a^{6} b^{9}} - \sqrt[5]{\frac{32}{243} a^{10} b^{15}} + \sqrt[3]{a^{6} b^{9}}$.
Printed answer:
- $\frac{3}{2}a^2b^3$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem Rational(3,2)*a**2*b**3
Exercise 27, problem 18
$ \sqrt[3]{\frac{1}{8} x^{6} y^{3}} + \sqrt[5]{-\frac{1}{32} x^{10} y^{5}} - \sqrt[3]{-x^{6} y^{3}} - \sqrt{\frac{1}{4} x^{4} y^{2}} $.
Printed answer:
- $\frac{1}{2}x^2y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(1,2)*x**2*y
Exercise 27, problem 19
Multiply $\sqrt{25 a^{4} b^{2} c^{2}}$ by $-\sqrt[3]{-8 a^{3} b^{6} c^{9}}$.
Printed answer:
- $10a^3b^3c^4$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: FLAG-DOMAIN 10*a**3*b**3*c**4
Exercise 27, problem 2
$\sqrt[4]{81 x^{4} y^{8}}$.
Printed answer:
- $3xy^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x*y**2
Exercise 27, problem 20
Divide $\sqrt{144 x^{4} y^{4}}$ by $\sqrt[5]{243 x^{10} y^{5}}$.
Printed answer:
- $4y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*y
Exercise 27, problem 21
Multiply $-\sqrt[3]{-125 x^{3} y^{6} z^{6}}$ by $\sqrt{9 x^{4} y^{2} z^{6}}$.
Printed answer:
- $15x^3y^3z^5$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 15*x**3*y**3*z**5
Exercise 27, problem 22
Divide $\sqrt{100 a^{6} b^{12}}$ by $\sqrt[5]{32 a^{15} b^{25}}$.
Printed answer:
- $5b$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*b
Exercise 27, problem 23
From two cities $a$ miles apart two trains start toward each other, the one going $x$ miles an hour, and the other $y$ miles an hour. How long before they will meet? How far will each train have gone?
Printed answer:
- $\displaystyle \frac{a}{x + y}$ hrs.; $\displaystyle \frac{ax}{x + y}$ miles.; $\displaystyle \frac{ay}{x + y}$ miles.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {t: a/(x + y), d1: a*x/(x + y), d2: a*y/(x + y)}
Exercise 27, problem 24
What number is that whose double exceeds its half by 39?
Printed answer:
- $26$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {n: 26}
On the STU-32 (STU, rpn):
```
39 ENTER 2 ENTER 2 1/x −
```
```
÷
```
Calculator: +26E+0; the book prints 26. Run on the calculator core at firmware 628c96c.
Exercise 27, problem 25
Mr. A. is $m$ years old, and 6 years ago he was one-half as old as Mr. B. How old was Mr. B. then? How old is Mr. B. now?
Printed answer:
- $2(m - 6)$; $2m - 6$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {b_then: 2*(m - 6), b_now: 2*m - 6}
Exercise 27, problem 3
$\sqrt[3]{-8 x^{6} y^{3}}$.
Printed answer:
- $- 2x^2y$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: FLAG-DOMAIN -2*x**2*y
Exercise 27, problem 4
$\sqrt[5]{-32 a^{10} b^{15}}$.
Printed answer:
- $- 2a^2b^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -2*a**2*b**3
Exercise 27, problem 5
$\sqrt[5]{243 a^{5} b^{10}}$.
Printed answer:
- $3ab^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*a*b**2
Exercise 27, problem 6
$\sqrt[3]{27 x^{3} y^{9}}$.
Printed answer:
- $3xy^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x*y**3
Exercise 27, problem 7
$\sqrt[4]{16 x^{8} y^{4}}$.
Printed answer:
- $2x^2y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**2*y
Exercise 27, problem 8
$\sqrt{9 x^{8} y^{6}}$.
Printed answer:
- $3x^4y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x**4*y**3
Exercise 27, problem 9
$\sqrt{\frac{4}{9} m^{6} y^{4}}$.
Printed answer:
- $\frac{2}{3} m^3y^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(2,3)*m**3*y**2

Exercise 28

Exercise 28, problem 1
$4x^2-12xy+9y^2$.
Printed answer:
- $2x - 3y$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*x - 3*y
Exercise 28, problem 10
$x^6+2x^5-x^4+3x^2-2x+1$.
Printed answer:
- $x^3 + x^2 - x + 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**3 + x**2 - x + 1
Exercise 28, problem 11
$6x^4-4x^3+4x^5-15x^2-8x+x^6+16$.
Printed answer:
- $x^3 + 2x^2 + x - 4$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**3 + 2*x**2 + x - 4
Exercise 28, problem 12
$7x^2-6x-11x^4+10x^3-4x^5+1+4x^6$.
Printed answer:
- $2x^3 - x^2 - 3x + 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*x**3 - x**2 - 3*x + 1
Exercise 28, problem 13
A is $x$ years old. If his age is as much above 50 as B’s age is below 40, what is B’s age?
Printed answer:
- $90 - x$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 90 - x
Exercise 28, problem 14
If $x$ represents the first digit, and $y$ the second digit, of a number, what will represent the number?
Printed answer:
- $10x + y$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 10*x + y
Exercise 28, problem 2
$x^4+10x^3y^3+25x^2y^6$.
Printed answer:
- $x^2 + 5xy^3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**2 + 5*x*y**3
Exercise 28, problem 3
$16a^2b^2c^4-56abc^2xy^2z+49x^2y^4z^2$.
Printed answer:
- $4abc^2 - 7xy^2z$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 4*a*b*c**2 - 7*x*y**2*z
Exercise 28, problem 4
$\frac{1}{4}x^2-xy^2z+y^4z^2$.
Printed answer:
- $\frac{1}{2} x - y^2z$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x/2 - y**2*z
Exercise 28, problem 5
$a^2b^6+\frac{2}{3}ab^3c^4+\frac{1}{9}c^8$.
Printed answer:
- $ab^3 + \frac{1}{3}c^4$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles a*b**3 + c**4/3
Exercise 28, problem 6
$x^4-4x^3+2x^2+4x+1$.
Printed answer:
- $x^2 - 2x - 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**2 - 2*x - 1
Exercise 28, problem 7
$x^4+6x^3+17x^2+24x+16$.
Printed answer:
- $x^2 + 3x + 4$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**2 + 3*x + 4
Exercise 28, problem 8
$4x^4+9x^2+4-4x-4x^3$.
Printed answer:
- $2x^2 - x + 2$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*x**2 - x + 2
Exercise 28, problem 9
$6x^3-5x^2+1-2x+9x^4$.
Printed answer:
- $3x^2 + x - 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 3*x**2 + x - 1

Exercise 29

Exercise 29, problem 1
$27 x^{3} - 27 x^{2} y + 9 x y^2 - y^{3}$.
Printed answer:
- $3x-y$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 3*x - y
Exercise 29, problem 10
$x^{9} - 7 x^{6} + x^{3} - 3 x^{4} - 3 x^{8} + 6 x^{7} + 6 x^{5}$.
Printed answer:
- $x^3-x^2+x$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**3 - x**2 + x
Exercise 29, problem 11
$5 x^{3} - 3 x - 3 x^{5} - 1 + x^{6}$.
Printed answer:
- $x^2-x-1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**2 - x - 1
Exercise 29, problem 12
$\frac{1}{8} a^{6} + \frac{3}{2} a^{5} + 5 \frac{1}{4} a^{4} + 2 a^{3} - 10 \frac{1}{2} a^{2} + 6 a - 1$.
Printed answer:
- $\frac{1}{2}a^2+2a-1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles Rational(1, 2)*a**2 + 2*a - 1
Exercise 29, problem 13
A board is $8 x$ inches long and $2 x$ inches wide. What is the length of a square board having the same area?
Printed answer:
- $4x$ in.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {s: 4*x}
Exercise 29, problem 14
If $3 x$ represents the edge of a cubical box, what represents the cubical contents of the box?
Printed answer:
- $27x^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes, with the problem read into an equation 27*x**3
Exercise 29, problem 15
If a cubical cistern contains $64y^3$ cubic feet, how long is one edge?
Printed answer:
- $4y$ ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {s: 4*y}
Exercise 29, problem 16a
A man travels north $a$ miles, and then south $b$ miles. How far is he from the starting-point? How far has he traveled?
Printed answer:
- $a-b$ miles north; $a+b$ miles.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles a - b
Exercise 29, problem 16b
A man travels north $a$ miles, and then south $b$ miles. How far is he from the starting-point? How far has he traveled?
Printed answer:
- $a-b$ miles north; $a+b$ miles.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles a + b
Exercise 29, problem 2
$15 x^{2} - 1 - 75 x^{4} + 125 x^{6}$.
Printed answer:
- $5x^2-1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 5*x**2 - 1
Exercise 29, problem 3
$144 a^{2} b^{2} + 27 b^{6} + 108 a b^{4} + 64 a^{3}$.
Printed answer:
- $3b^2+4a$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 3*b**2 + 4*a
Exercise 29, problem 4
$x^{6} - 8 y^{6} + 12 x^{2} y^{4} - 6 x^{4} y^{2}$.
Printed answer:
- $x^2-2y^2$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**2 - 2*y**2
Exercise 29, problem 5
$1 + 9 x + 27 x^{2} + 27 x{3}$.
Printed answer:
- $1+3x$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 1 + 3*x
Exercise 29, problem 6
$1 - 21 m - 343 m^{3} + 147 m^{2}$.
Printed answer:
- $1-7m$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 1 - 7*m
Exercise 29, problem 7
$3 x^{2} + 64 x^{6} - 24 x^{4} - \frac{1}{8}$.
Printed answer:
- $4x^2-\frac{1}{2}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 4*x**2 - Rational(1, 2)
Exercise 29, problem 8
$27 x^{9} + \frac{1}{27} + x^{3} + 9 x^{6}$.
Printed answer:
- $3x^3+\frac{1}{3}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 3*x**3 + Rational(1, 3)
Exercise 29, problem 9
$8 a^{6} + 18 a^{4} + 9 a^{2} - 12 a^{5} - 13 a^{3} - 3 a + 1$.
Printed answer:
- $2a^2-a+1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*a**2 - a + 1

Exercise 30

Exercise 30, problem 1
$\sqrt{2025}$.
Printed answer:
- $45$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 45
On the STU-32 (STU, rpn):
```
2025 √x
```
Calculator: +45E+0; the book prints 45. Run on the calculator core at firmware 628c96c.
Exercise 30, problem 10
$\sqrt[3]{389017}$.
Printed answer:
- $73$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 73
On the STU-32 (STU, rpn):
```
389017 ENTER 3 GOLD ˣ√y
```
Calculator: +730E-1; the book prints 73. Run on the calculator core at firmware 628c96c.
Exercise 30, problem 11
$\sqrt[3]{241804.367}$.
Printed answer:
- $62.3$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 62.3
On the STU-32 (STU, rpn):
```
241804.367 ENTER 3 GOLD ˣ√y
```
Calculator: +6230E-2; the book prints 62.3. Run on the calculator core at firmware 628c96c.
Exercise 30, problem 12
$\sqrt{7039.21}$.
Printed answer:
- $83.9$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 83.9
On the STU-32 (STU, rpn):
```
7039.21 √x
```
Calculator: +839E-1; the book prints 83.9. Run on the calculator core at firmware 628c96c.
Exercise 30, problem 13
$\sqrt[3]{35.287552}$.
Printed answer:
- $3.28$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 3.28
On the STU-32 (STU, rpn):
```
35.287552 ENTER 3 GOLD ˣ√y
```
Calculator: +3280E-3; the book prints 3.28. Run on the calculator core at firmware 628c96c.
Exercise 30, problem 14
$\sqrt{2550.25}$.
Printed answer:
- $50.5$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 50.5
Exercise 30, problem 15
$\sqrt{34.78312}$ to three decimal places.
Printed answer:
- $5.898-$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 5.898
Exercise 30, problem 16
$\sqrt{7}$ to three decimal places.
Printed answer:
- $2.646-$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 2.646
Exercise 30, problem 17
$\sqrt[3]{.1255}$ to three decimal places.
Printed answer:
- $.501-$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(501, 1000)
Exercise 30, problem 18
A merchant bought a bale of cloth containing just as many pieces as there were yards in each piece. The whole number of yards was 1089. What was the number of pieces?
Printed answer:
- $33$ pieces.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 33}
On the STU-32 (STU, rpn):
```
1089 √x
```
Calculator: +33E+0; the book prints 33. Run on the calculator core at firmware 628c96c.
Exercise 30, problem 19
A regiment, consisting of 5476 men, is to be formed into a solid square. How many men must be placed in each rank?
Printed answer:
- $74$ men.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 74}
On the STU-32 (STU, rpn):
```
5476 √x
```
Calculator: +74E+0; the book prints 74. Run on the calculator core at firmware 628c96c.
Exercise 30, problem 2
$\sqrt{9409}$.
Printed answer:
- $97$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 97
Exercise 30, problem 20
What is the depth of a cubical cistern which contains 5000 gallons of water? (1 gallon = 231 cubic inches.)
Printed answer:
- $104.9+$ in.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 104.9}
Exercise 30, problem 21
A farmer plants an orchard containing 8464 trees, and has as many rows of trees as there are trees in each row. What is the number of trees in each row?
Printed answer:
- $92$ trees.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 92}
On the STU-32 (STU, rpn):
```
8464 √x
```
Calculator: +92E+0; the book prints 92. Run on the calculator core at firmware 628c96c.
Exercise 30, problem 3
$\sqrt{20449}$.
Printed answer:
- $143$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 143
Exercise 30, problem 4
$\sqrt{904401}$.
Printed answer:
- $951$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 951
Exercise 30, problem 5
$\sqrt{70.56}$.
Printed answer:
- $8.4$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 8.4
On the STU-32 (STU, rpn):
```
70.56 √x
```
Calculator: +84E-1; the book prints 8.4. Run on the calculator core at firmware 628c96c.
Exercise 30, problem 6
$\sqrt{.9025}$.
Printed answer:
- $.95$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 0.95
Exercise 30, problem 7
$\sqrt{94864}$.
Printed answer:
- $308$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 308
Exercise 30, problem 8
$\sqrt{.00000784}$
Printed answer:
- $.0028$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 0.0028
Exercise 30, problem 9
$\sqrt[3]{59.319}$.
Printed answer:
- $3.9$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 3.9
On the STU-32 (STU, rpn):
```
59.319 ENTER 3 1/x yˣ
```
Calculator: +3899999999999999999999999999999999E-33; the book prints 3.9. Run on the calculator core at firmware 628c96c.