Positive and Negative Numbers

Excerpts

Positive and Negative Numbers
If, now, we wish to subtract $7$ from $4$, we begin at $4$ in the positive series, count $7$ units in the *negative direction* (to the left), and arrive at $-3$ in the negative series; that is, $4 - 7 = -3$.
Positive and Negative Numbers
If we wish to subtract $5$ from $2$, we cannot do it, because when we have counted backwards from $2$ as far as $0$, *the natural series of numbers comes to an end*.
Positive and Negative Numbers
In order to subtract a greater number from a smaller, it is necessary to *assume* a new series of numbers, beginning at zero and extending to the left of zero.
Positive and Negative Numbers
When no sign stands before a number, the sign $+$ is always understood. Thus $4$ means the same as $+4$, $a$ means the same as $+a$. But *the sign $-$ is never omitted*.
Positive and Negative Numbers
In the first sense they are signs of *operations*, and are common to Arithmetic and Algebra; in the second sense they are signs of *opposition*, and are employed in Algebra alone.
Positive and Negative Numbers
From these four cases it follows that in finding the product of two algebraic numbers, if the two numbers have *like* signs, the product will have the *plus* sign, and if *unlike* signs, the product will have the *minus* sign.

Equations

Positive and Negative Numbers
+a - a = 0
Adding a number to its opposite (equal absolute value, unlike sign) gives zero, so +a and -a cancel.
Positive and Negative Numbers
a^{m} × a^{n} = a^{m + n}
The product of two powers of the same number has the index equal to the sum of the indices of the factors.
Positive and Negative Numbers
(+a) × (+b) &= +ab
The product of two positive numbers is positive (like signs give plus).
Positive and Negative Numbers
(+a) × (-b) &= -ab
A positive number times a negative number is negative (unlike signs give minus).
Positive and Negative Numbers
(-a) × (+b) &= -ab
A negative number times a positive number is negative (unlike signs give minus).
Positive and Negative Numbers
(-a) × (-b) &= +ab
The product of two negative numbers is positive (like signs give plus).
Positive and Negative Numbers
+ab ÷ (+a) = +b
Dividing a positive product by a positive factor gives a positive quotient (like signs give plus).
Positive and Negative Numbers
-ab ÷ (+a) = -b
Dividing a negative dividend by a positive divisor gives a negative quotient (unlike signs give minus).
Positive and Negative Numbers
-ab ÷ (-a) = +b
Dividing a negative dividend by a negative divisor gives a positive quotient (like signs give plus).
Positive and Negative Numbers
+ab ÷ (-a) = -b
Dividing a positive dividend by a negative divisor gives a negative quotient (unlike signs give minus).
Positive and Negative Numbers
a^{m} - a^{n} = a^{m - n}
FLAG: as transcribed, the chapter prints a minus sign between the two powers, but the surrounding text (the index law in division, §78, and the worked quotients a^5/a^2 = a^(5-2) etc.) shows that the intended operation is division, a^m ÷ a^n = a^(m-n); the printed sign is most likely a transcription error in the source. Recorded verbatim here, not corrected; the sympy field follows the printed text. For m greater than n, the index of the quotient of two powers of the same number is m minus n.

Problems

Exercise 15

Exercise 15, problem 1, p. 39
$5c$, $23c$, $c$, $11c$.
Printed answer:
- $40c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 40*c
Exercise 15, problem 10, p. 39
$5d$, $-d$, $-4d$, $2d$.
Printed answer:
- $2d$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*d
Exercise 15, problem 11, p. 39
$13n$, $13n$, $-11n$, $-6n$, $-9n$, $n$, $2n$, $-3n$.
Printed answer:
- $0$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 0
Exercise 15, problem 12, p. 39
$5g$, $-3g$, $-6g$, $-4g$, $20g$, $-5g$, $-11g$, $-14g$.
Printed answer:
- $-18g$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -18*g
Exercise 15, problem 13, p. 39
$-9a^{2}$, $5a^{2}$, $6a^{2}$, $a^{2}$, $2a^{2}$, $-a^{2}$, $-3a^{2}$.
Printed answer:
- $a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2
Exercise 15, problem 14, p. 39
$3x^{3}$, $-2x^{3}$, $-5x^{3}$, $-7x^{3}$, $-x^{3}$, $2x^{3}$, $-10x^{3}$, $-x^{3}$.
Printed answer:
- $-21x^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -21*x**3
Exercise 15, problem 15, p. 39
$4a^{2}b^{2}$, $-a^{2}b^{2}$, $-6a^{2}b^{2}$, $4a^{2}b^{2}$, $-2a^{2}b^{2}$, $a^{2}b^{2}$.
Printed answer:
- $0$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 0
Exercise 15, problem 16, p. 39
$6mn$, $-5mn$, $mn$, $-3mn$, $4mn$.
Printed answer:
- $3mn$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*m*n
Exercise 15, problem 17, p. 39
$3xyz$, $-2xyz$, $5xyz$, $-7xyz$, $xyz$.
Printed answer:
- $0$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 0
Exercise 15, problem 18, p. 39
$5a^{3}b^{3}c^{3}$, $-7a^{3}b^{3}c^{3}$, $-3a^{3}b^{3}c^{3}$, $2a^{3}b^{3}c^{3}$.
Printed answer:
- $-3a^{3}b^{3}c^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -3*a**3*b**3*c**3
Exercise 15, problem 19, p. 39
$11abcd$, $-10abcd$, $-9abcd$, $-abcd$.
Printed answer:
- $-9abcd$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -9*a*b*c*d
Exercise 15, problem 2, p. 39
$4a$, $3a$, $7a$, $10a$.
Printed answer:
- $24a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 24*a
Exercise 15, problem 20, p. 39
Subtract $-a$ from $-b$, and find the value of the result if $a = -4$, $b = -5$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 1
On the STU-32 (STU, rpn):
```
5 +/− +/− ENTER
```
```
4 +/− +/− −
```
Calculator: +1E+0; the book prints 1. Run on the calculator core at firmware 628c96c.
Exercise 15, problem 21, p. 39
$a - b + c$ and $-a + b + c$.
Printed answer:
- $12$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 12
Exercise 15, problem 22, p. 39
$a + (-b) + c$ and $a - (-b) + c$.
Printed answer:
- $4$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 4
On the STU-32 (STU, rpn):
```
4 ENTER 2 +/− +/− + 3 +/− +
```
```
4 ENTER 2 +/− +/− − 3 +/− +
```
```
−
```
Calculator: +4E+0; the book prints 4. Run on the calculator core at firmware 628c96c.
Exercise 15, problem 23, p. 39
$-a - (-b) + c$ and $-(-a) + (-b) - c$.
Printed answer:
- $-18$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes -18
Exercise 15, problem 24, p. 39
$a - b + (-c)$ and $a - (-b) - (-c)$.
Printed answer:
- $10$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 10
On the STU-32 (STU, rpn):
```
4 ENTER 2 +/− − 3 +
```
```
4 ENTER 2 − 3 −
```
```
−
```
Calculator: +10E+0; the book prints 10. Run on the calculator core at firmware 628c96c.
Exercise 15, problem 3, p. 39
$7x$, $12x$, $11x$, $9x$.
Printed answer:
- $39x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 39*x
Exercise 15, problem 4, p. 39
$6y$, $8y$, $2y$, $35y$.
Printed answer:
- $51y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 51*y
Exercise 15, problem 5, p. 39
$-3a$, $-5a$, $-18a$.
Printed answer:
- $-26a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -26*a
Exercise 15, problem 6, p. 39
$-5x$, $-6x$, $-18x$, $-11x$.
Printed answer:
- $-40x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -40*x
Exercise 15, problem 7, p. 39
$-3b$, $-b$, $-9b$, $-4b$.
Printed answer:
- $-17b$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -17*b
Exercise 15, problem 8, p. 39
$-z$, $-2z$, $-10z$, $-53z$.
Printed answer:
- $-66z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -66*z
Exercise 15, problem 9, p. 39
$-11m$, $-3m$, $-5m$, $-m$.
Printed answer:
- $-20m$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -20*m

Exercise 16

Exercise 16, problem 1, p. 42
$5a^{2}$ and $6a^{3}$.
Printed answer:
- $30a^{5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 30*a**5
Exercise 16, problem 10, p. 42
$-2x^{6}y^{3}z$ and $-6xy^{2}z$.
Printed answer:
- $12x^{7}y^{5}z^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 12*x**7*y**5*z**2
Exercise 16, problem 11, p. 42
$3a^{2}b$, $-5ab^{2}$, and $-7a^{4}b^{2}$.
Printed answer:
- $105a^{7}b^{5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 105*a**7*b**5
Exercise 16, problem 12, p. 42
$2a^{2}bc^{3}$, $-3a^{3}b^{2}c$, and $-ab^{2}c^{3}$.
Printed answer:
- $6a^{6}b^{5}c^{7}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 6*a**6*b**5*c**7
Exercise 16, problem 13, p. 42
$2b^{2}c^{2}x^{2}$, $2a^{2}b^{2}c^{3}$, and $-3a^{3}bx^{3}$.
Printed answer:
- $-12a^{5}b^{5}c^{5}x^{5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -12*a**5*b**5*c**5*x**5
Exercise 16, problem 14, p. 42
$2a^{3}b^{2}c$, $-3a^{2}b^{3}c$, and $-4a^{2}bc^{3}$.
Printed answer:
- $24a^{7}b^{6}c^{5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 24*a**7*b**6*c**5
Exercise 16, problem 15, p. 42
$7am^{2}x^{3}$, $3a^{4}m^{2}x^{3}$, and $-2amx$.
Printed answer:
- $-42a^{6}m^{5}x^{7}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -42*a**6*m**5*x**7
Exercise 16, problem 16, p. 42
$-3x^{2}y^{2}z^{2}$, $2x^{2}yz^{3}$, and $-5x^{4}yz$.
Printed answer:
- $30x^{8}y^{4}z^{6}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 30*x**8*y**4*z**6
Exercise 16, problem 17, p. 42
$2ab^{2} - 3bc^{2} + c$.
Printed answer:
- $-46$
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
2 ENTER 2 +/− ×
```
```
3 GOLD x² ×
```
```
3 ENTER 3 × 1 +/− GOLD x² × −
```
```
1 +/− +
```
Calculator: -46E+0; the book prints -46. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 18, p. 42
$4a^{2} - 2b^{2} - c^{2}$.
Printed answer:
- $-3$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
4 ENTER 2 +/− GOLD x² ×
```
```
2 ENTER 3 GOLD x² × −
```
```
1 +/− GOLD x² −
```
Calculator: -3E+0; the book prints -3. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 19, p. 42
$5a + 2b - 4c^{4}$.
Printed answer:
- $-8$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
5 ENTER 2 +/− ×
```
```
2 ENTER 3 × +
```
```
1 +/− ENTER 4 yˣ 4 × −
```
Calculator: -8E+0; the book prints -8. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 2, p. 42
$8ab$ and $5a^{3}b^{2}$.
Printed answer:
- $40a^{4}b^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 40*a**4*b**3
Exercise 16, problem 20, p. 42
$2a^{3} - 3b + 8c^{2}$.
Printed answer:
- $-17$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
2 +/− ENTER 3 yˣ 2 ×
```
```
3 ENTER 3 × −
```
```
1 +/− GOLD x² 8 × +
```
Calculator: -17E+0; the book prints -17. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 21, p. 42
$-a + 3b - 2c^{2}$.
Printed answer:
- $9$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
3 ENTER 3 ×
```
```
2 ENTER 1 +/− GOLD x² ×
```
```
−
```
```
2 +/− +/−
```
```
+
```
Calculator: +9E+0; the book prints 9. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 22, p. 42
$-a^{3} - 2b - 10c$.
Printed answer:
- $12$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
2 +/− ENTER 3 yˣ +/−
```
```
2 ENTER 3 × −
```
```
10 ENTER 1 +/− × −
```
Calculator: +12E+0; the book prints 12. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 23, p. 42
$3a^{3} - 3b^{3} - 3c^{3}$.
Printed answer:
- $-102$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
2 +/− ENTER 3 yˣ 3 ×
```
```
3 ENTER 3 yˣ 3 × −
```
```
1 +/− ENTER 3 yˣ 3 × −
```
Calculator: -102E+0; the book prints -102. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 24, p. 42
$2ab^{2} - 3bc^{2} + 2ac$.
Printed answer:
- $-41$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
2 ENTER 2 +/− × 3 GOLD x² ×
```
```
3 ENTER 3 × 1 +/− GOLD x² × −
```
```
2 ENTER 2 +/− × 1 +/− × +
```
Calculator: -41E+0; the book prints -41. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 25, p. 42
$3abc + 5a^{2}b^{2} - 2a^{2}b$.
Printed answer:
- $174$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
3 ENTER 2 +/− × 3 × 1 +/− ×
```
```
5 ENTER 2 +/− GOLD x² × 3 GOLD x² × +
```
```
2 ENTER 2 +/− GOLD x² × 3 × −
```
Calculator: +174E+0; the book prints 174. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 26, p. 42
$ab^{2}c^{2} + 2abc^{2} + a^{2}b^{2}c^{2}$.
Printed answer:
- $6$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
2 +/− ENTER 3 GOLD x² ×
```
```
1 +/− GOLD x² ×
```
```
2 ENTER 2 +/− × 3 × 1 +/− GOLD x² ×
```
```
+
```
```
2 +/− GOLD x² 3 GOLD x² × 1 +/− GOLD x² × +
```
Calculator: +6E+0; the book prints 6. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 27, p. 42
$2a^{2}bc + 3abc + a^{2}b^{2}c^{2}$.
Printed answer:
- $30$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
2 ENTER 2 +/− GOLD x² × 3 × 1 +/− ×
```
```
3 ENTER 2 +/− × 3 × 1 +/− ×
```
```
+
```
```
2 +/− GOLD x² 3 GOLD x² × 1 +/− GOLD x² × +
```
Calculator: +30E+0; the book prints 30. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 28, p. 42
$6a^{2} + 8a^{2}b^{2} - 5a^{2}bc$.
Printed answer:
- $372$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes
On the STU-32 (STU, rpn):
```
6 ENTER 2 +/− GOLD x² ×
```
```
8 ENTER 2 +/− GOLD x² × 3 GOLD x² × +
```
```
5 ENTER 2 +/− GOLD x² × 3 × 1 +/− × −
```
Calculator: +372E+0; the book prints 372. Run on the calculator core at firmware 628c96c.
Exercise 16, problem 3, p. 42
$9xy$ and $7xy$.
Printed answer:
- $63x^{2}y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 63*x**2*y**2
Exercise 16, problem 4, p. 42
$2a^{2}b$ and $a^{3}b^{4}c^{2}$.
Printed answer:
- $2a^{5}b^{5}c^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a**5*b**5*c**2
Exercise 16, problem 5, p. 42
$3a^{3}b^{7}c^{8}$ and $3a^{4}b^{2}c$.
Printed answer:
- $9a^{7}b^{9}c^{9}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*a**7*b**9*c**9
Exercise 16, problem 6, p. 42
$2a$ and $-5a$.
Printed answer:
- $-10a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -10*a**2
Exercise 16, problem 7, p. 42
$-3a$ and $-4b$.
Printed answer:
- $12ab$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 12*a*b
Exercise 16, problem 8, p. 42
$-ab$ and $a^{3}b^{2}$.
Printed answer:
- $-a^{4}b^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -a**4*b**3
Exercise 16, problem 9, p. 42
$-2ab^{4}$ and $-5a^{4}bc$.
Printed answer:
- $10a^{5}b^{5}c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 10*a**5*b**5*c

Exercise 17

Exercise 17, problem 1, p. 45
$x^{3}$ by $x$.
Printed answer:
- $x^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2
Exercise 17, problem 10, p. 45
$-25x^{4}y^{2}$ by $-5x^{3}y^{2}$.
Printed answer:
- $5x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*x
Exercise 17, problem 11, p. 45
$-51x^{2}y^{3}$ by $-17x^{2}y$.
Printed answer:
- $3y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*y**2
Exercise 17, problem 12, p. 45
$-28a^{4}b^{3}$ by $7a^{3}b$.
Printed answer:
- $-4ab^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -4*a*b**2
Exercise 17, problem 13, p. 45
$-36x^{2}y^{6}$ by $-3xy^{2}$.
Printed answer:
- $12xy^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 12*x*y**4
Exercise 17, problem 14, p. 45
$-3x^{4}y^{6}$ by $-5xy^{3}$.
Printed answer:
- $\dfrac{3x^{3}y^{3}}{5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x**3*y**3/5
Exercise 17, problem 15, p. 45
$-12a^{2}b^{3}$ by $8ab^{3}$.
Printed answer:
- $-\dfrac{3a}{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -3*a/2
Exercise 17, problem 16, p. 45
$-abcd$ by $ac$.
Printed answer:
- $-bd$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -b*d
Exercise 17, problem 17, p. 45
$-a^{2}b^{3}c^{4}d^{5}$ by $-ab^{3}c^{3}d^{3}$.
Printed answer:
- $acd^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*c*d**2
Exercise 17, problem 18, p. 45
$2x^{2}y^{2}z^{3}$ by $-3xyz^{3}$.
Printed answer:
- $-\dfrac{2xy}{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -2*x*y/3
Exercise 17, problem 19, p. 45
$-5a^{5}b^{3}c^{7}$ by $-a^{4}b^{2}c^{7}$.
Printed answer:
- $5ab$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*a*b
Exercise 17, problem 2, p. 45
$21x^{5}$ by $7x^{3}$.
Printed answer:
- $3x^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x**2
Exercise 17, problem 20, p. 45
$52a^{2}m^{3}n^{4}$ by $13a^{2}m^{2}n^{3}$.
Printed answer:
- $4mn$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*m*n
Exercise 17, problem 21, p. 45
$13xy^{2}z^{4}$ by $39xyz$.
Printed answer:
- $\dfrac{yz^{3}}{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y*z**3/3
Exercise 17, problem 22, p. 45
$68xc^{2}d^{3}$ by $-4xcd^{2}$.
Printed answer:
- $-17cd$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -17*c*d
Exercise 17, problem 23, p. 45
$-8m^{5}n^{3}p^{2}$ by $-4m^{5}np$.
Printed answer:
- $2n^{2}p$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*n**2*p
Exercise 17, problem 24, p. 45
$-6pqr^{3}$ by $-2p^{2}qr$.
Printed answer:
- $\dfrac{3r^{2}}{p}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*r**2/p
Exercise 17, problem 25, p. 45
$26a^{2}g^{2}t^{5}$ by $-2agt^{4}$.
Printed answer:
- $-13agt$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -13*a*g*t
Exercise 17, problem 26, p. 45
$-a^{4}b^{2}c^{3}$ by $-a^{5}b^{3}c^{4}$.
Printed answer:
- $\dfrac{1}{abc}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1/(a*b*c)
Exercise 17, problem 27, p. 45
$-3x^{2}y^{2}z^{2}$ by $-2x^{3}y^{4}z^{5}$.
Printed answer:
- $\dfrac{3}{2xy^{2}z^{3}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3/(2*x*y**2*z**3)
Exercise 17, problem 28, p. 45
$-6mnp$ by $-3m^{2}n^{2}p^{2}$.
Printed answer:
- $\dfrac{2}{mnp}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2/(m*n*p)
Exercise 17, problem 29, p. 45
$-17a^{2}b^{3}c^{4}$ by $51ab^{5}c^{4}$.
Printed answer:
- $-\dfrac{a}{3b^{2}}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -a/(3*b**2)
Exercise 17, problem 3, p. 45
$35x^{2}$ by $-5x^{2}$.
Printed answer:
- $-7$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -7
Exercise 17, problem 30, p. 45
$-19mg^{2}t^{3}$ by $57m2^gt^{4}$.
Printed answer:
- $-\dfrac{g}{3mt}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem -g/(3*m*t)
Exercise 17, problem 4, p. 45
$-42x^{2}$ by $6x^{2}$.
Printed answer:
- $-7$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -7
Exercise 17, problem 5, p. 45
$-63x^{5}$ by $-9x$.
Printed answer:
- $7x^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 7*x**4
Exercise 17, problem 6, p. 45
$-72x^{3}$ by $-8x^{2}$.
Printed answer:
- $9x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*x
Exercise 17, problem 7, p. 45
$-32a^{2}b^{2}$ by $8ab^{2}$.
Printed answer:
- $-4a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -4*a
Exercise 17, problem 8, p. 45
$-16x^{3}y^{3}$ by $-4xy$.
Printed answer:
- $4x^{2}y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2*y**2
Exercise 17, problem 9, p. 45
$18x^{2}y$ by $-2xy$.
Printed answer:
- $-9x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -9*x