MULTIPLICATION

Excerpts

MULTIPLICATION
In arithmetic you learned that multiplication is the addition of equal numbers, that the multiplicand expresses one of those equal numbers, and the multiplier the number of them.
MULTIPLICATION
This must mean the opposite of what $+4$ meant as a multiplier. Plus four meant add, minus four must mean subtract.
MULTIPLICATION
*To multiply a monomial by a monomial, multiply the coefficients together for the coefficient of the product, add the exponents of like letters for the exponent of the same letter in the product, and give the product of two numbers having like signs the plus sign, having unlike signs the minus sign.*
MULTIPLICATION
To multiply a polynomial by a polynomial, multiply the multiplicand by each term of the multiplier, and add the products.
MULTIPLICATION
It is intended that the answers in the following exercise shall be written directly without going through the multiplication.

Equations

MULTIPLICATION
a^3 = a \cdot a \cdot a
The third power of a is a times a times a.
MULTIPLICATION
(x-y)^4=x^4-4x^3y+6x^2y^2-4xy^3+y^4
The fourth power of x minus y expands to x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4, written directly without multiplying out.
MULTIPLICATION
(x-1)^3=x^3-3x^2+3x-1
The cube of x minus 1 expands to x^3 - 3x^2 + 3x - 1.

Problems

Exercise 18

Exercise 18, problem 1
$5x$ and $7c$.
Printed answer:
- $35 cx$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 35*c*x
Exercise 18, problem 10
$-\frac{2}{3} a^{2}b^{2}$, $\frac{1}{4} c^{2}$, $-\frac{6}{5}ac$, and $-\frac{3}{4} b^{3}c$.
Printed answer:
- $-\frac{3}{20}a^3b^5c^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -Rational(3, 20)*a**3*b**5*c**4
Exercise 18, problem 11
In how many days will $a$ boys eat 100 apples if each boy eats $b$ apples a day?
Printed answer:
- $\frac{100}{ab}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 100/(a*b)
Exercise 18, problem 12
How many units in $x$ hundreds?
Printed answer:
- $100x$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 100*x
Exercise 18, problem 13
If there are $a$ hundreds, $b$ tens, and $c$ units in a number, what will represent the whole number of units?
Printed answer:
- $100a + 10b + c$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 100*a + 10*b + c
Exercise 18, problem 14
If the difference between two numbers is 7, and one of the numbers is $x$, what is the other number?
Printed answer:
- $x + 7$ or $x - 7$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation [x + 7, x - 7]
Exercise 18, problem 2
$51cy$ and $-xa$.
Printed answer:
- $-51 acxy$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -51*a*c*x*y
Exercise 18, problem 3
$3x^{3}y$ and $7axy^{2}$.
Printed answer:
- $21ax^4y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 21*a*x**4*y**3
Exercise 18, problem 4
$5a^{2}bc$ and $2ab^{2}c^{3}$.
Printed answer:
- $10a^3b^3c^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 10*a**3*b**3*c**4
Exercise 18, problem 5
$-3x^{2}y$, $-2ax$, and $3cy^{2}$.
Printed answer:
- $18acx^3y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 18*a*c*x**3*y**3
Exercise 18, problem 6
$5a^{2}$, $-3bc^{3}$, and $-2abc$.
Printed answer:
- $30a^3b^2c^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 30*a**3*b**2*c**4
Exercise 18, problem 7
$15x^{2}y^{2}$, $-\frac{2}{3} xz$, and $\frac{1}{10} yz^{2}$.
Printed answer:
- $-x^3y^3z^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -x**3*y**3*z**3
Exercise 18, problem 8
$20a^{3}b^{2}$, $\frac{2}{5} ab^{2}$, and $-\frac{1}{8} bc^{2}$.
Printed answer:
- $-a^4b^5c^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -a**4*b**5*c**2
Exercise 18, problem 9
$\frac{1}{2} xy$, $-\frac{2}{3} cx^{2}$, $-\frac{2}{5} a^{2}y$, and $-\frac{5}{3} x^{2}y^{2}$.
Printed answer:
- $-\frac{2}{9}a^2cx^6y^4$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem -Rational(2, 9)*a**2*c*x**6*y**4

Exercise 19

Exercise 19, problem 1
$ x^2 + xy + y^2 \mbox{by} x^2y^2 $.
Printed answer:
- $x^4y^2 + x^3y^3 + x^2y^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4*y**2 + x**3*y**3 + x**2*y**4
Exercise 19, problem 10
$x^4 - x^3 + x^2 - x + 1$ by $2 + 3x + 2x^2 + x^3$.
Printed answer:
- $x^7 + x^6 + 2x^5 + x^2 + x + 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**7 + x**6 + 2*x**5 + x**2 + x + 2
Exercise 19, problem 11
$a^5 - a^4b + a^3b^2 - a^2b^3 + ab^4 - b^5$ by $a + b$.
Printed answer:
- $a^6 + b^6$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem a**6 + b**6
Exercise 19, problem 12
$x^2 - xy + y^2 - yz + z^2 - xz$ by $x + y + z$.
Printed answer:
- $x^3 - 3xyz + y^3 + z^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 - 3*x*y*z + y**3 + z**3
Exercise 19, problem 13
$x^6 + x^5y + x^4y^2 + x^3y^3 + x^2y^4 + xy^5 + y^6$ by $x - y$.
Printed answer:
- $x^7 - y^7$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**7 - y**7
Exercise 19, problem 14
$x^4 - 4a^2x^2 + 4a^4$ by $x^4 + 4a^2x^2 + 4a^4$.
Printed answer:
- $x^8 - 8x^4a^4 + 16a^8$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**8 - 8*x**4*a**4 + 16*a**8
Exercise 19, problem 15
$a^3 - 3a^2y^2 + y^3$ by $a^3 + 3a^2y^2 +y^3$.
Printed answer:
- $a^6 + 2a^3y^3 - 9a^4y^4 + y^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**6 + 2*a**3*y**3 - 9*a**4*y**4 + y**6
Exercise 19, problem 16
$x^4 + 10x + 12 + 9x^2 + 3x^3$ by $-2x + x^2 - 1$.
Printed answer:
- $x^6 + x^5 + 2x^4 - 11x^3 - 17x^2 - 34x - 12$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**6 + x**5 + 2*x**4 - 11*x**3 - 17*x**2 - 34*x - 12
Exercise 19, problem 17
$3x^2 - 2 + x^3 - 3x + 6x^4$ by $-2 + x^2 - 3x$.
Printed answer:
- $6x^6 - 17x^5 - 12x^4 - 14x^3 + x^2 + 12x + 4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 6*x**6 - 17*x**5 - 12*x**4 - 14*x**3 + x**2 + 12*x + 4
Exercise 19, problem 18a
If $x$ represent the number of miles a man can row in an hour in still water, how far can the man row in 5 hours down a stream which flows $y$ miles an hour? How far up the same stream in 4 hours?
Printed answer:
- $5(x + y)$
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 5*(x + y)
Exercise 19, problem 18b
If $x$ represent the number of miles a man can row in an hour in still water, how far can the man row in 5 hours down a stream which flows $y$ miles an hour? How far up the same stream in 4 hours?
Printed answer:
- $4(x - y)$
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 4*(x - y)
Exercise 19, problem 19
A can reap a field in 7 hours, and B can reap the same field in 5 hours. How much of the field can they do in one hour, working together?
Printed answer:
- $\frac{12}{35}$ of the field.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
Exercise 19, problem 2
$ a^2 - ab + b^2 \mbox{by} a^2b $.
Printed answer:
- $a^4b - a^3b^2 +a^2b^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4*b - a**3*b**2 + a**2*b**3
Exercise 19, problem 20
A tank can be filled by two pipes in $a$ hours and $b$ hours respectively. What part of the tank will be filled by both pipes running together for one hour?
Printed answer:
- $\frac{1}{a} + \frac{1}{b}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 1/a + 1/b
Exercise 19, problem 3
$ a^3 - 3a^2b + b^3 \mbox{by} -2ab $.
Printed answer:
- $-2a^4b + 6a^3b^2 - 2ab^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -2*a**4*b + 6*a**3*b**2 - 2*a*b**4
Exercise 19, problem 4
$ 8x^3 + 36x^2y + 27y^3 by 3xy^2 $.
Printed answer:
- $24x^4y^2 + 108x^3y^3 + 81xy^5$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 24*x**4*y**2 + 108*x**3*y**3 + 81*x*y**5
Exercise 19, problem 5
$ \frac{5}{6}a^4 - \frac{1}{5}a^3b - \frac{1}{3}a^2b^2 \mbox{by} \frac{6}{5}ab^2 $.
Printed answer:
- $a^5b^2 - \frac{6}{25}a^4b^3 - \frac{2}{5}a^3b^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**5*b**2 - Rational(6, 25)*a**4*b**3 - Rational(2, 5)*a**3*b**4
Exercise 19, problem 6
$ x^2 - xy + y^2 \mbox{by} x + y $.
Printed answer:
- $x^3 + y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 + y**3
Exercise 19, problem 7
$x^4 - 3x^3 + 2x^2 - x + 1$ by $x-1$.
Printed answer:
- $x^5 - 4x^4 + 5x^3 - 3x^2 + 2x - 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**5 - 4*x**4 + 5*x**3 - 3*x**2 + 2*x - 1
Exercise 19, problem 8
$x^3 - 2x^2 + x$ by $x^2 + 3x + 1$.
Printed answer:
- $x^5 + x^4 - 4x^3 + x^2 + x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**5 + x**4 - 4*x**3 + x**2 + x
Exercise 19, problem 9
$xy + mn - xm - yn$ by $xy - mn + xm - yn$.
Printed answer:
- $x^2y^2 - 2xy^2n + y^2n^2 - m^2n^2 + 2xm^2n - x^2m^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2*y**2 - 2*x*y**2*n + y**2*n**2 - m**2*n**2 + 2*x*m**2*n - x**2*m**2

Exercise 20

Exercise 20, problem 1
$(x + 2)(x + 7)$.
Printed answer:
- $x^2 + 9x + 14$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 9*x + 14
Exercise 20, problem 10
$(x - 13)(x - 1)$.
Printed answer:
- $x^2 - 14x + 13$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 14*x + 13
Exercise 20, problem 11
$(y + 7)(y - 9)$.
Printed answer:
- $y^2 - 2y - 63$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**2 - 2*y - 63
Exercise 20, problem 12
$(x + 3)(x + 17)$.
Printed answer:
- $x^2 + 20x + 51$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 20*x + 51
Exercise 20, problem 13
$(y+2)(y-15)$.
Printed answer:
- $y^2 - 13y - 30$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**2 - 13*y - 30
Exercise 20, problem 14
$(y+2)(y+16)$.
Printed answer:
- $y^2 + 18y + 32$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**2 + 18*y + 32
Exercise 20, problem 15
$(a^2 + 7)(a^2 - 5)$.
Printed answer:
- $a^4 + 2a^2 - 35$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4 + 2*a**2 - 35
Exercise 20, problem 16
$(a-9)(a+9)$.
Printed answer:
- $a^2 - 81$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 - 81
Exercise 20, problem 17
$(m^2 - 2)(m^2 - 16)$.
Printed answer:
- $m^4 - 18m^2 + 32$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m**4 - 18*m**2 + 32
Exercise 20, problem 18
$(b^3 + 12)(b^3 - 10)$.
Printed answer:
- $b^6 + 2b^3 - 120$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes b**6 + 2*b**3 - 120
Exercise 20, problem 19
$(x - \frac{1}{2})(x - \frac{1}{4})$.
Printed answer:
- $x^2 - \frac{3}{4}x + \frac{1}{8}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - Rational(3, 4)*x + Rational(1, 8)
Exercise 20, problem 2
$(x + 1)(x + 6)$.
Printed answer:
- $x^2 + 7x + 6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 7*x + 6
Exercise 20, problem 20
$(y + \frac{1}{3})(y + \frac{1}{6})$.
Printed answer:
- $y^2 + \frac{1}{2}y + \frac{1}{18}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**2 + Rational(1, 2)*y + Rational(1, 18)
Exercise 20, problem 21
$(m + \frac{2}{3})(m - \frac{1}{3})$.
Printed answer:
- $m^2 + \frac{1}{3}m - \frac{2}{9}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m**2 + Rational(1, 3)*m - Rational(2, 9)
Exercise 20, problem 22
$(a - \frac{2}{5})(a + \frac{3}{5})$.
Printed answer:
- $a^2 + \frac{1}{5}a - \frac{6}{25}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 + Rational(1, 5)*a - Rational(6, 25)
Exercise 20, problem 23
$(x - \frac{2}{3})(x - \frac{1}{2})$.
Printed answer:
- $x^2 - \frac{7}{6}x + \frac{1}{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - Rational(7, 6)*x + Rational(1, 3)
Exercise 20, problem 24
$(y + \frac{3}{4})(y + \frac{1}{5})$.
Printed answer:
- $y^2 + \frac{19}{20}y + \frac{3}{20}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**2 + Rational(19, 20)*y + Rational(3, 20)
Exercise 20, problem 25
$(3-x)(7-x)$.
Printed answer:
- $21 - 10x + x^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 21 - 10*x + x**2
Exercise 20, problem 26
$(5-x)(3-x)$.
Printed answer:
- $15 -8x + x^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 15 - 8*x + x**2
Exercise 20, problem 27
$(6-x)(7+x)$.
Printed answer:
- $42 - x - x^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 42 - x - x**2
Exercise 20, problem 28
$(11-x)(3+x)$.
Printed answer:
- $33 + 8x - x^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 33 + 8*x - x**2
Exercise 20, problem 29
$(x-3)(x+3)$.
Printed answer:
- $x^2 - 9$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 9
Exercise 20, problem 3
$(x - 3)(x - 4)$.
Printed answer:
- $x^2 - 7x + 12$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 7*x + 12
Exercise 20, problem 30
$(y+5)(y-5)$.
Printed answer:
- $y^2 - 25$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**2 - 25
Exercise 20, problem 31
Find a number which, being multiplied by 6, and having 15 added to the product, will equal 141.
Printed answer:
- $21$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 21
On the STU-32 (STU, rpn):
```
141 ENTER 15 −
```
```
6 ÷
```
Calculator: +21E+0; the book prints 21. Run on the calculator core at firmware 628c96c.
Exercise 20, problem 32
Mr. Allen has 3 more cows than his neighbor. Three times his number of cows will equal four times his neighbor’s. How many has Mr. Allen?
Printed answer:
- $12$ cows.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {a: 12}
On the STU-32 (STU, rpn):
```
3 ENTER 3 ×
```
```
3 +
```
Calculator: +12E+0; the book prints 12. Run on the calculator core at firmware 628c96c.
Exercise 20, problem 4
$(x - 5)(x - 2)$.
Printed answer:
- $x^2 - 7x + 10$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 7*x + 10
Exercise 20, problem 5
$(x + 5)(x - 2)$.
Printed answer:
- $x^2 + 3x - 10$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 3*x - 10
Exercise 20, problem 6
$(x + 7)(x - 3)$.
Printed answer:
- $x^2 + 4x - 21$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 4*x - 21
Exercise 20, problem 7
$(x - 7)(x + 6)$.
Printed answer:
- $x^2 - x - 42$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - x - 42
Exercise 20, problem 8
$(x - 6)(x + 5)$.
Printed answer:
- $x^2 - x - 30$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - x - 30
Exercise 20, problem 9
$(x - 11)(x - 2)$.
Printed answer:
- $x^2 - 13x + 22$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 13*x + 22

Exercise 21

Exercise 21, problem 1
$(a^2b)^2$.
Printed answer:
- $a^4b^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4*b**2
Exercise 21, problem 10
$(11c^5d^12x^4)^2$.
Printed answer:
- $121 c^{10}d^{24}x^8$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 121*c**10*d**24*x**8
Exercise 21, problem 11
$(\frac{1}{2}x^2am^3)^2$.
Printed answer:
- $\frac{1}{4} x^4a^2m^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(1, 4)*x**4*a**2*m**6
Exercise 21, problem 12
$(-\frac{1}{3}ab^3c)^2$.
Printed answer:
- $\frac{1}{9} a^2b^6c^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(1, 9)*a**2*b**6*c**2
Exercise 21, problem 13
$(-15c^6dx^2)^2$.
Printed answer:
- $225 c^{12}d^2x^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 225*c**12*d**2*x**4
Exercise 21, problem 14
$(-9xy^5z^2)^3$.
Printed answer:
- $-729 x^3y^{15}z^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -729*x**3*y**15*z**6
Exercise 21, problem 15
$(a^9b^2c^4d^2)^4$.
Printed answer:
- $a^{36}b^8c^{16}d^8$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**36*b**8*c**16*d**8
Exercise 21, problem 16
$(-x^8yz^3m^2n)^5$.
Printed answer:
- $-x^{40}y^5z^{15}m^{10}n^5$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -x**40*y**5*z**15*m**10*n**5
Exercise 21, problem 17
$(-\frac{2}{3}a^2bc^4)^2$.
Printed answer:
- $\frac{4}{9} a^4b^2c^8$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(4, 9)*a**4*b**2*c**8
Exercise 21, problem 18
$(\frac{5}{6}mn^2x^3)^2$.
Printed answer:
- $\frac{25}{36} m^2n^4x^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(25, 36)*m**2*n**4*x**6
Exercise 21, problem 19
In how many days can one man do as much as $b$ men in 8 days?
Printed answer:
- $8 b$ days.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {d: 8*b}
Exercise 21, problem 2
$(xy^2)^3$.
Printed answer:
- $x^3y^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3*y**6
Exercise 21, problem 20a
How many mills in $a$ cents? How many dollars?
Printed answer:
- $10 a$ mills; $\frac{a}{100}$ dols.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 10*a
Exercise 21, problem 20b
How many mills in $a$ cents? How many dollars?
Printed answer:
- $10 a$ mills; $\frac{a}{100}$ dols.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes a/100
Exercise 21, problem 3
$(-a^4b)^2$.
Printed answer:
- $a^8b^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**8*b**2
Exercise 21, problem 4
$(-x^3y^2)^3$.
Printed answer:
- $-x^9y^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -x**9*y**6
Exercise 21, problem 5
$(3a^2y)^3$.
Printed answer:
- $27 a^6y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 27*a**6*y**3
Exercise 21, problem 6
$(-7ab^2c^3)^2$.
Printed answer:
- $49 a^2b^4c^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 49*a**2*b**4*c**6
Exercise 21, problem 7
$(xyz^2)^5$.
Printed answer:
- $x^5y^5z^{10}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**5*y**5*z**10
Exercise 21, problem 8
$(-m^2nd)^4$.
Printed answer:
- $m^8n^4d^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m**8*n**4*d**4
Exercise 21, problem 9
$(-5x^3y^4z)^3$.
Printed answer:
- $-125 x^9y^{12}z^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -125*x**9*y**12*z**3

Exercise 22

Exercise 22, problem 1
$(z+x)^3$.
Printed answer:
- $z^3 + 3z^2x + 3zx^2 + x^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes z**3 + 3*z**2*x + 3*z*x**2 + x**3
Exercise 22, problem 10
$(y^2+z^4)^3$.
Printed answer:
- $y^6 + 3y^4z^4 + 3y^2z^8 + z^{12}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**6 + 3*y**4*z**4 + 3*y**2*z**8 + z**12
Exercise 22, problem 11
$(x^2y+z)^2$.
Printed answer:
- $x^4y^2 + 2x^2yz + z^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4*y**2 + 2*x**2*y*z + z**2
Exercise 22, problem 12
$(a^2b-c)^4$.
Printed answer:
- $a^8b^4 - 4a^6b^3c + 6a^4b^2c^2-4a^2bc^3 + c^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**8*b**4 - 4*a**6*b**3*c + 6*a**4*b**2*c**2 - 4*a**2*b*c**3 + c**4
Exercise 22, problem 13
$(a^2-b^3c)^3$.
Printed answer:
- $a^6 - 3a^4b^3c + 3a^2b^6c^2-b^9c^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**6 - 3*a**4*b**3*c + 3*a**2*b**6*c**2 - b**9*c**3
Exercise 22, problem 14
$(x^2y-mn^3)^2$.
Printed answer:
- $x^4y^2 - 2x^2ymn^3 + m^2n^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4*y**2 - 2*x**2*y*m*n**3 + m**2*n**6
Exercise 22, problem 15
$(x+1)^3$.
Printed answer:
- $x^3 + 3x^2 + 3x + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 + 3*x**2 + 3*x + 1
Exercise 22, problem 16
$(m-1)^2$.
Printed answer:
- $m^2 - 2m + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m**2 - 2*m + 1
Exercise 22, problem 17
$(b^2-1)^4$.
Printed answer:
- $b^8 - 4b^6 + 6b^4 - 4b^2 + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes b**8 - 4*b**6 + 6*b**4 - 4*b**2 + 1
Exercise 22, problem 18
$(y^3+1)^3$.
Printed answer:
- $y^9 + 3y^6 + 3y^3 + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**9 + 3*y**6 + 3*y**3 + 1
Exercise 22, problem 19
$(ab-2)^2$.
Printed answer:
- $a^2b^2 - 4ab + 4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2*b**2 - 4*a*b + 4
Exercise 22, problem 2
$(a+y)^4$.
Printed answer:
- $a^4 + 4a^3y + 6a^2y^2 + 4ay^3 + y^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4 + 4*a**3*y + 6*a**2*y**2 + 4*a*y**3 + y**4
Exercise 22, problem 20
$(x^2y-3)^2$.
Printed answer:
- $x^4y^2 - 6x^2y + 9$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4*y**2 - 6*x**2*y + 9
Exercise 22, problem 21
$(1-x)^4$.
Printed answer:
- $1 - 4x + 6x^2 - 4x^3 + x^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 - 4*x + 6*x**2 - 4*x**3 + x**4
Exercise 22, problem 22
$(1-y^2)^3$.
Printed answer:
- $1 - 3y^2 + 3y^4 - y^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 - 3*y**2 + 3*y**4 - y**6
Exercise 22, problem 23
$(2x+3y^2)^2$.
Printed answer:
- $4x^2 + 12xy^2 + 9y^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2 + 12*x*y**2 + 9*y**4
Exercise 22, problem 24
$(3ab-x^2y)^3$.
Printed answer:
- $27a^3b^3 - 27a^2b^2x^2y + 9abx^4y^2 - x^6y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 27*a**3*b**3 - 27*a**2*b**2*x**2*y + 9*a*b*x**4*y**2 - x**6*y**3
Exercise 22, problem 25
$(4mn^3-3a^2b)^4$.
Printed answer:
- $256m^4n^{12} - 768m^3n^9a^2b + 864m^2n^6a^4b^2 - 432mn^3a^6b^3 + 81a^8b^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 256*m**4*n**12 - 768*m**3*n**9*a**2*b + 864*m**2*n**6*a**4*b**2 - 432*m*n**3*a**6*b**3 + 81*a**8*b**4
Exercise 22, problem 26
$(\frac{1}{2}x-y)^2$.
Printed answer:
- $\frac{1}{4}x^2 - xy + y^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(1,4)*x**2 - x*y + y**2
Exercise 22, problem 27
$(1-\frac{1}{3}x^2)^3$.
Printed answer:
- $1 - x^2 + \frac{1}{3}x^4 - \frac{1}{27}x^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 - x**2 + Rational(1,3)*x**4 - Rational(1,27)*x**6
Exercise 22, problem 28
$(x^2-3)^4$.
Printed answer:
- $x^8 - 12x^6 + 54x^4 - 108x^2 + 81$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**8 - 12*x**6 + 54*x**4 - 108*x**2 + 81
Exercise 22, problem 29
John has $4a$ horses, James has $a$ times as many as John, and Charles has $d$ less than five times as many as James. How many has Charles?
Printed answer:
- $20a^2 - d$ horses.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {C: 20*a**2 - d}
Exercise 22, problem 3
$(x-a)^4$.
Printed answer:
- $x^4 - 4x^3a + 6x^2a^2 - 4xa^3 + a^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4 - 4*x**3*a + 6*x**2*a**2 - 4*x*a**3 + a**4
Exercise 22, problem 30
A man bought $a$ pounds of meat at $a$ cents a pound, and handed the butcher an $x$-dollar bill. How many cents in change should he receive?
Printed answer:
- $100x - a^2$ cts.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {M: a**2, B: 100*x, C: 100*x - a**2}
Exercise 22, problem 31
A grocer, having 25 bags of meal worth $a$ cents a bag, sold $x$ bags. What is the value of the meal left?
Printed answer:
- $a(25 - x)$ cts.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {V: a*(25 - x)}
Exercise 22, problem 32
If $a = 5$, $x = 4$, $y = 3$, find the numerical value of 7a11x-3y + 11x8x-7y - 10y7a-5x.
Printed answer:
- $3$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 3
Exercise 22, problem 33
Find the value of a^2b - c^2d - (ab+cd)(ac-bd) - bc(a^2c-bd^2) when $a = 2$, $b = 3$, $c = 4$, and $d = 0$.
Printed answer:
- $-228$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes -228
Exercise 22, problem 4
$(a-m)^3$.
Printed answer:
- $a^3 - 3a^2m + 3am^2 - m^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**3 - 3*a**2*m + 3*a*m**2 - m**3
Exercise 22, problem 5
$(m+a)^2$.
Printed answer:
- $m^2 + 2am + a^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m**2 + 2*a*m + a**2
Exercise 22, problem 6
$(x-y)^2$.
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 22, problem 7
$(x^2+y^2)^3$.
Printed answer:
- $x^6 + 3x^4y^2 + 3x^2y^4 + y^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**6 + 3*x**4*y**2 + 3*x**2*y**4 + y**6
Exercise 22, problem 8
$(m^3-y^2)^2$.
Printed answer:
- $m^6 - 2m^3y^2 + y^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m**6 - 2*m**3*y**2 + y**4
Exercise 22, problem 9
$(c^2-d^2)^4$.
Printed answer:
- $c^8 - 4c^6d^2 + 6c^4d^4 - 4c^2d^6 + d^8$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes c**8 - 4*c**6*d**2 + 6*c**4*d**4 - 4*c**2*d**6 + d**8

Exercise 23

Exercise 23, problem 1
Take the sum of $x^3 + 3x - 2$, $2x^3 + x^2 - x + 5$, and $4x^3 + 2x^2 - 7x + 4$ from the sum of $2x^3 + 9x$ and $5x^3 + 3x^2$.
Printed answer:
- $14x - 7$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 14*x - 7
Exercise 23, problem 10
Simplify $(x-2)(x+7) + (x-8)(x-5)$.
Printed answer:
- $2x^2 - 8x + 26$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**2 - 8*x + 26
Exercise 23, problem 11
Expand $(2a^2b-3xy)^3$.
Printed answer:
- $8a^6 b^3 - 36a^4 b^2 xy + 54a^2 bx^2 y^2 - 27x^3 y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 8*a**6*b**3 - 36*a**4*b**2*x*y + 54*a**2*b*x**2*y**2 - 27*x**3*y**3
Exercise 23, problem 12
What must be subtracted from $x^3-3x^2+2y-5$ to produce unity?
Printed answer:
- $x^3 - 3x^2 + 2y - 6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 - 3*x**2 + 2*y - 6
Exercise 23, problem 13
Multiply $x^3+ 3x^2y+3xy^2 + y^3$ by $3xy^2 - y^3-3x^2y+x^3$.
Printed answer:
- $x^6 - 3x^4 y^2 + 3x^2 y^4 - y^6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**6 - 3*x**4*y**2 + 3*x**2*y**4 - y**6
Exercise 23, problem 14
Expand $(x+1)(x-1)(x^2+1)$.
Printed answer:
- $x^4 - 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4 - 1
Exercise 23, problem 15
Add $4xy^3-4y^4$, $4x^3y-12x^2y^2+12xy^3-4y^4$, $6x^2y^2-12xy^3+6y^4$ and $x^4-4x^3y+6x^2y^2-4xy^3+y^4$.
Printed answer:
- $x^4 - y^4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4 - y**4
Exercise 23, problem 16
A man weighs 36 pounds more than his wife, and the sum of their weights is 317 pounds. What is the weight of each?
Printed answer:
- $176\frac{1}{2}$ lbs.; $140\frac{1}{2}$ lbs.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {m: 353/2, w: 281/2}
Exercise 23, problem 17
A watch and chain cost $350. What was the cost of each, if the chain cost $\frac{3}{4}$ as much as the watch?
Printed answer:
- watch, $200; chain, $150.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {w: 200, c: 150}
Exercise 23, problem 18
Simplify $3x^2-2x+1-(x^2+2x+3)-(2x^2-6x-6)$.
Printed answer:
- $2x + 4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x + 4
Exercise 23, problem 19
Simplify $(a + 2y)^2-(a-2y)^2$
Printed answer:
- $8\,ay$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 8*a*y
Exercise 23, problem 2
Multiply $b^4 - 2b^2$ by $b^4 + 2b^2 - 1$.
Printed answer:
- $b^8 - 2b^4 + 1$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem b**8 - 2*b**4 + 1
Exercise 23, problem 20
What is the value of $1 + \frac{1}{2}a + \frac{1}{3}b$ times $1- \frac{1}{2}a + \frac{1}{3}b$ ?
Printed answer:
- $1 + \frac{2}{3} b + \frac{1}{9} b^2 - \frac{1}{4} a^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 + Rational(2,3)*b + Rational(1,9)*b**2 - Rational(1,4)*a**2
Exercise 23, problem 3
Simplify $11x^2 + 4y^2 - (2xy - 3y^2) + (2x^2 - 3xy) - (3x^2 - 5xy)$.
Printed answer:
- $10x^2 + 7y^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 10*x**2 + 7*y**2
Exercise 23, problem 4
Divide $\$300$ among $A$, $B$, and $C$, so that $A$ shall have twice as much as $B$, and $B$ $$20$ more than $C$.
Printed answer:
- $160; $80; $60.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {A: 160, B: 80, C: 60}
Exercise 23, problem 5
Find two numbers differing by $8$ such that four times the less may exceed twice the greater by $10$.
Printed answer:
- 13; 21.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 13, y: 21}
Exercise 23, problem 6
What must be added to $3a^3 - 4a^2 - 4$ to produce $5a^3 + 6$?
Printed answer:
- $2a^3 + 4a^2 + 10$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a**3 + 4*a**2 + 10
Exercise 23, problem 7
Add $\frac{2}{3}a^2 - ab - \frac{5}{4}b^2$, $\frac{2}{3}a^2 + \frac{1}{3}ab - \frac{1}{4}b^2$, and $-a^2 -\frac{2}{3}ab + 2b^2$.
Printed answer:
- $\frac{1}{3} a^2 - \frac{4}{3} ab + \frac{1}{2} b^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(1,3)*a**2 - Rational(4,3)*a*b + Rational(1,2)*b**2
Exercise 23, problem 8
Simplify $8ab^2c^4 \times (-3a^4bc^2) \times (-2a^2b^3c)$.
Printed answer:
- $48a^7 b^6 c^7$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 48*a**7*b**6*c**7
Exercise 23, problem 9
Expand $\left(-\frac{2}{3}xy^2z^3\right)^4$.
Printed answer:
- $\frac{16}{81}x^4 y^8 z^{12}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes Rational(16,81)*x**4*y**8*z**12