2. (a) what is the multiplicative inverse of ⅓? (b) what is the multiplicative inverse of -⅓? (c) what is the product of any nonzero real number and its multiplicative inverse? 3. what is another name for the multiplicative inverse of a number? 4. which real number does not have a multiplicative inverse and why? 5. use two unit multipliers to convert 112 feet to centimeters. 6. use two unit multipliers to convert 60 square kilometers to square meters. 7. the area of a rectangle is 72 square centimeters. the length of one side is 6 centimeters. what is the length of the other side? 8. the length of the base of a right triangle is 12 meters. the height of the right triangle is 9 meters. find the area of the right triangle. 9. the diameter of a circle is 16 kilometers. find the area of the circle. evaluate: 10. a - ab if a = 2 and b = -3 11. xy - 3y if x = 2 and y = 4 12. 2ab - 3abc if a = -1, b = 2, and c = 3 13. -x(a - b) if x = -2, a = 3, and b = -1 14. -x - (-a + b) if x = 3, a = -2, and b = 4 15. (x - y)(y - x) if x = -2 and y = 3 16. (-x) + (-y) if x = -3 and y = 2 17. -c - (p - c) if p = -5 and c = 2 18. (a - x)(ma - x) if a = -3, x = -4, and m = 5 simplify: 19. -2(-6 - 1 - 2) - (-2 + 7) 20. -|-11| + (-3)|-3 + 5| 21. -3(-3)(-2 - 5 + |-11|) 22. -3{(-5 - 2)(-1)} 23. $\frac{-4 - (-1 - 3)}{-7 - (-9 + 2)}$ 24. $\frac{3(-5 + 2)+6(-4 + 10)-2}{(10 - 2)+7 + 5}$ 25. find the perimeter of this figure. all angles are right angles. dimensions are in inches. 26. find the area of this figure. corners that look square are square. dimensions are in feet.

Question

Accepted Answer

### 2 (a)
# Explanation:
## Step1: Recall multiplicative - inverse definition
The multiplicative inverse of a non - zero number $a$ is $\frac{1}{a}$ such that $a	imes\frac{1}{a}=1$. For $a = \frac{1}{3}$, its multiplicative inverse is $3$ since $\frac{1}{3}	imes3 = 1$.
# Answer:
$3$

### 2 (b)
# Explanation:
## Step1: Apply multiplicative - inverse rule
The multiplicative inverse of a non - zero number $a$ is $\frac{1}{a}$. For $a=-\frac{1}{7}$, its multiplicative inverse is $- 7$ because $-\frac{1}{7}	imes(-7)=1$.
# Answer:
$-7$

### 2 (c)
# Explanation:
## Step1: Use multiplicative - inverse property
By the definition of multiplicative inverse, for any non - zero real number $x$, its multiplicative inverse is $\frac{1}{x}$, and $x	imes\frac{1}{x}=1$.
# Answer:
$1$

### 3
# Explanation:
## Step1: Recall alternative name
The multiplicative inverse of a number is also called the reciprocal of the number.
# Answer:
Reciprocal

### 4
# Explanation:
## Step1: Consider the property of multiplicative inverse
The real number $0$ does not have a multiplicative inverse because if we assume the multiplicative inverse of $0$ is $x$, then $0	imes x = 1$. But the product of $0$ and any real number is $0$, not $1$.
# Answer:
The real number $0$ does not have a multiplicative inverse because the equation $0	imes x = 1$ has no solution for $x\in\mathbb{R}$.

### 5
# Explanation:
## Step1: Know the conversion factors
We know that $1$ foot $ = 12$ inches and $1$ inch $=2.54$ centimeters.
## Step2: Set up unit - multipliers
$112$ feet $	imes\frac{12	ext{ inches}}{1	ext{ foot}}	imes\frac{2.54	ext{ centimeters}}{1	ext{ inch}}$.
## Step3: Calculate
$112	imes12	imes2.54=112	imes30.48 = 3413.76$ centimeters.
# Answer:
$3413.76$ centimeters

### 6
# Explanation:
## Step1: Recall area - conversion factors
$1$ kilometer $ = 1000$ meters, so $1$ square kilometer $=(1000)^2$ square meters.
## Step2: Set up unit - multipliers
$60$ square kilometers $	imes\frac{1000	ext{ meters}}{1	ext{ kilometer}}	imes\frac{1000	ext{ meters}}{1	ext{ kilometer}}$.
## Step3: Calculate
$60	imes1000	imes1000 = 60000000$ square meters.
# Answer:
$60000000$ square meters

### 7
# Explanation:
## Step1: Use the area formula for a rectangle
The area of a rectangle is $A = l	imes w$. Given $A = 72$ square centimeters and $l = 6$ centimeters. We solve for $w$ using $w=\frac{A}{l}$.
## Step2: Calculate
$w=\frac{72}{6}=12$ centimeters.
# Answer:
$12$ centimeters

### 8
# Explanation:
## Step1: Recall the area formula for a right - triangle
The area of a right - triangle is $A=\frac{1}{2}	imes base	imes height$. Given base $b = 12$ meters and height $h = 9$ meters.
## Step2: Calculate
$A=\frac{1}{2}	imes12	imes9=54$ square meters.
# Answer:
$54$ square meters

### 9
# Explanation:
## Step1: Find the radius of the circle
Given the diameter $d = 16$ kilometers, the radius $r=\frac{d}{2}=8$ kilometers.
## Step2: Use the area formula for a circle
The area of a circle is $A=\pi r^{2}$. Substitute $r = 8$ kilometers into the formula: $A=\pi	imes(8)^{2}=64\pi$ square kilometers.
# Answer:
$64\pi$ square kilometers

### 10
# Explanation:
## Step1: Substitute the values of $a$ and $b$
Given $a = 2$ and $b=-3$, substitute into $a - ab$. We get $2-2	imes(-3)$.
## Step2: Perform the multiplication first
$2-(-6)$.
## Step3: Simplify
$2 + 6=8$.
# Answer:
$8$

### 11
# Explanation:
## Step1: Substitute the values of $x$ and $y$
Given $x = 2$ and $y = 4$, substitute into $xy-3y$. We get $2	imes4-3	imes4$.
## Step2: Perform the multiplications
$8 - 12=-4$.
# Answer:
$-4$

### 12
# Explanation:
## Step1: Substitute the values of $a$, $b$, and $c$
Given $a=-1$, $b = 2$, and $c = 3$, substitute into $2ab-3abc$. We get $2	imes(-1)	imes2-3	imes(-1)	imes2	imes3$.
## Step2: Perform the multiplications
$-4 + 18=14$.
# Answer:
$14$

### 13
# Explanation:
## Step1: Substitute the values of $x$, $a$, and $b$
Given $x=-2$, $a = 3$, and $b=-1$, substitute into $-x(a - b)$. We get $-(-2)(3-(-1))$.
## Step2: Simplify the expression inside the parentheses
$2	imes(3 + 1)$.
## Step3: Perform the multiplication
$2	imes4 = 8$.
# Answer:
$8$

### 14
# Explanation:
## Step1: Substitute the values of $x$, $a$, and $b$
Given $x = 3$, $a=-2$, and $b = 4$, substitute into $-x-(-a + b)$. We get $-3-(-(-2)+4)$.
## Step2: Simplify the inner - parentheses
$-3-(2 + 4)$.
## Step3: Simplify further
$-3-6=-9$.
# Answer:
$-9$

### 15
# Explanation:
## Step1: Substitute the values of $x$ and $y$
Given $x=-2$ and $y = 3$, substitute into $(x - y)(y - x)$. We get $(-2-3)(3-(-2))$.
## Step2: Simplify the expressions inside the parentheses
$(-5)	imes(3 + 2)$.
## Step3: Perform the multiplication
$(-5)	imes5=-25$.
# Answer:
$-25$

### 16
# Explanation:
## Step1: Substitute the values of $x$ and $y$
Given $x=-3$ and $y = 2$, substitute into $(-x)+(-y)$. We get $-(-3)+(-2)$.
## Step2: Simplify
$3-2 = 1$.
# Answer:
$1$

### 17
# Explanation:
## Step1: Substitute the values of $p$ and $c$
Given $p=-5$ and $c = 2$, substitute into $-c-(p - c)$. We get $-2-(-5 - 2)$.
## Step2: Simplify the inner - parentheses
$-2-(-7)$.
## Step3: Simplify further
$-2 + 7=5$.
# Answer:
$5$

### 18
# Explanation:
## Step1: Substitute the values of $a$, $x$, and $m$
Given $a=-3$, $x=-4$, and $m = 5$, substitute into $(a - x)(ma - x)$. We get $(-3-(-4))(5	imes(-3)-(-4))$.
## Step2: Simplify the expressions inside the parentheses
$( - 3 + 4)(-15 + 4)$.
## Step3: Perform the multiplications
$1	imes(-11)=-11$.
# Answer:
$-11$

### 19
# Explanation:
## Step1: Simplify inside the first parentheses
$-2(-6-1 - 2)-(-2 + 7)=-2(-9)-5$.
## Step2: Perform the multiplication
$18-5 = 13$.
# Answer:
$13$

### 20
# Explanation:
## Step1: Evaluate the absolute - value expressions
$-|-11|+(-3)|-3 + 5|=-11+(-3)	imes2$.
## Step2: Perform the multiplication
$-11-6=-17$.
# Answer:
$-17$

### 21
# Explanation:
## Step1: Simplify the expression inside the absolute - value and parentheses
$-3(-3)(-2-5+|-11|)=9(-2-5 + 11)$.
## Step2: Simplify the expression inside the parentheses
$9	imes4 = 36$.
# Answer:
$36$

### 22
# Explanation:
## Step1: Simplify the expression inside the innermost parentheses
$-3\{[(-5 - 2)](-1)\}=-3\{(-7)(-1)\}$.
## Step2: Perform the multiplication inside the braces
$-3	imes7=-21$.
# Answer:
$-21$

### 23
# Explanation:
## Step1: Simplify the numerator and denominator
The numerator: $-4-(-1-3)=-4-(-4)=-4 + 4 = 0$. The denominator: $-7-(-9 + 2)=-7-(-7)=-7 + 7 = 0$. Since the denominator is $0$, the fraction is undefined.
# Answer:
Undefined

### 24
# Explanation:
## Step1: Simplify the numerator and denominator
Numerator: $3(-5 + 2)+6(-4 + 10)-2=3	imes(-3)+6	imes6-2=-9 + 36-2=25$.
Denominator: $(10 - 2)+7 + 5=8 + 7+5=20$.
So the fraction is $\frac{25}{20}=\frac{5}{4}$.
# Answer:
$\frac{5}{4}$

### 25
# Explanation:
## Step1: Analyze the figure
The perimeter of the figure can be found by adding up the lengths of all the sides.
The outer rectangle has two sides of length $35$ inches and two sides of length $25$ inches. The inner "cut - out" part adds two sides of length $13$ inches.
$P=2	imes35+2	imes25+2	imes13=70 + 50+26=146$ inches.
# Answer:
$146$ inches

### 26
# Explanation:
## Step1: Split the figure into a rectangle and a semi - circle
The rectangle has length $7$ feet and width $5$ feet. The area of the rectangle is $A_{rect}=7	imes5 = 35$ square feet.
The semi - circle has a radius $r = 5$ feet. The area of a full - circle is $A_{circle}=\pi r^{2}=\pi	imes5^{2}=25\pi$ square feet, so the area of the semi - circle is $A_{semicircle}=\frac{1}{2}	imes25\pi=\frac{25\pi}{2}$ square feet.
The total area is $A = 35+\frac{25\pi}{2}$ square feet.
# Answer:
$35+\frac{25\pi}{2}$ square feet

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

2 (a)

Step1: Recall multiplicative - inverse definition

Step1: Apply multiplicative - inverse rule

Step1: Use multiplicative - inverse property

Answer:

2 (b)