simplify using the properties of exponents. circle your final answers.
1.) $\frac{42}{63}$
2.) $\frac{10^4}{10^6}$
3.) $\frac{12a}{3a}$
4.) $\frac{6 \cdot 10^5}{3 \cdot 10^3}$
5.) $\frac{4x^5}{8x}$
6.) $\frac{12y^3}{4xy}$
7.) $\frac{-9x^2y^3}{12xy^2}$
8.) $\frac{-28cd^3}{-21bd^2}$
9.) $\frac{xy^2z^3}{x^3y^2z}$
10.) $\frac{32a^2bc^3}{20abc}$
11.) $\frac{(2r)^4}{2r}$
12.) $\frac{(3x^2)^3}{(3x^3)^2}$
13.) $\frac{(2ab)^2}{2ab^2}$
14.) $\frac{(-x)^9}{(-x)^6}$
15.) $\frac{(-xy)^9}{xy^9}$

Question

Accepted Answer

### Problem 1: $\boldsymbol{\frac{42}{63}}$
# Explanation:
## Step1: Find GCD of 42 and 63
GCD of 42 and 63 is 21.
## Step2: Divide numerator and denominator by 21
$\frac{42\div21}{63\div21} = \frac{2}{3}$
# Answer: $\frac{2}{3}$

### Problem 2: $\boldsymbol{\frac{10^4}{10^6}}$
# Explanation:
## Step1: Use exponent rule $a^m\div a^n = a^{m - n}$
Here, $a = 10$, $m = 4$, $n = 6$. So, $10^{4-6}=10^{-2}$
## Step2: Simplify $10^{-2}$
$10^{-2}=\frac{1}{10^2}=\frac{1}{100}$
# Answer: $\frac{1}{100}$

### Problem 3: $\boldsymbol{\frac{12a}{3a}}$
# Explanation:
## Step1: Divide coefficients and variables separately
Divide 12 by 3, and $a$ by $a$.
## Step2: Simplify
$\frac{12}{3}	imes\frac{a}{a}=4	imes1 = 4$ (since $\frac{a}{a}=1$ for $a
eq0$)
# Answer: $4$

### Problem 4: $\boldsymbol{\frac{6\cdot10^5}{3\cdot10^3}}$
# Explanation:
## Step1: Divide coefficients and powers of 10 separately
$\frac{6}{3}	imes\frac{10^5}{10^3}$
## Step2: Simplify each part
$\frac{6}{3}=2$, and using $a^m\div a^n=a^{m - n}$, $\frac{10^5}{10^3}=10^{5 - 3}=10^2 = 100$
## Step3: Multiply results
$2	imes100 = 200$
# Answer: $200$

### Problem 5: $\boldsymbol{\frac{4x^5}{8x}}$
# Explanation:
## Step1: Divide coefficients and variables separately
$\frac{4}{8}	imes\frac{x^5}{x}$
## Step2: Simplify each part
$\frac{4}{8}=\frac{1}{2}$, and using $a^m\div a^n=a^{m - n}$, $\frac{x^5}{x}=x^{5 - 1}=x^4$
## Step3: Combine
$\frac{1}{2}x^4$
# Answer: $\frac{1}{2}x^4$

### Problem 6: $\boldsymbol{\frac{12y^3}{4xy}}$
# Explanation:
## Step1: Divide coefficients and variables separately
$\frac{12}{4}	imes\frac{y^3}{y}	imes\frac{1}{x}$
## Step2: Simplify each part
$\frac{12}{4}=3$, $\frac{y^3}{y}=y^{3 - 1}=y^2$, so we get $3	imes\frac{y^2}{x}=\frac{3y^2}{x}$
# Answer: $\frac{3y^2}{x}$

### Problem 7: $\boldsymbol{\frac{-9x^2y^3}{12xy^2}}$
# Explanation:
## Step1: Divide coefficients and variables separately
$\frac{-9}{12}	imes\frac{x^2}{x}	imes\frac{y^3}{y^2}$
## Step2: Simplify each part
$\frac{-9}{12}=-\frac{3}{4}$, $\frac{x^2}{x}=x^{2 - 1}=x$, $\frac{y^3}{y^2}=y^{3 - 2}=y$
## Step3: Combine
$-\frac{3}{4}xy$
# Answer: $-\frac{3}{4}xy$

### Problem 8: $\boldsymbol{\frac{-28cd^3}{-21bd^2}}$
# Explanation:
## Step1: Divide coefficients and variables separately
$\frac{-28}{-21}	imes\frac{c}{b}	imes\frac{d^3}{d^2}$
## Step2: Simplify each part
$\frac{-28}{-21}=\frac{4}{3}$, $\frac{d^3}{d^2}=d^{3 - 2}=d$, so we get $\frac{4}{3}	imes\frac{cd}{b}=\frac{4cd}{3b}$
# Answer: $\frac{4cd}{3b}$

### Problem 9: $\boldsymbol{\frac{xy^2z^3}{x^3y^2z}}$
# Explanation:
## Step1: Divide variables using exponent rule
For $x$: $x^1\div x^3=x^{1 - 3}=x^{-2}=\frac{1}{x^2}$; for $y$: $y^2\div y^2=y^{2 - 2}=y^0 = 1$; for $z$: $z^3\div z^1=z^{3 - 1}=z^2$
## Step2: Combine results
$\frac{1}{x^2}	imes1	imes z^2=\frac{z^2}{x^2}$
# Answer: $\frac{z^2}{x^2}$

### Problem 10: $\boldsymbol{\frac{32a^2bc^3}{20abc}}$
# Explanation:
## Step1: Divide coefficients and variables separately
$\frac{32}{20}	imes\frac{a^2}{a}	imes\frac{b}{b}	imes\frac{c^3}{c}$
## Step2: Simplify each part
$\frac{32}{20}=\frac{8}{5}$, $\frac{a^2}{a}=a^{2 - 1}=a$, $\frac{b}{b}=1$, $\frac{c^3}{c}=c^{3 - 1}=c^2$
## Step3: Combine
$\frac{8}{5}ac^2$
# Answer: $\frac{8}{5}ac^2$

### Problem 11: $\boldsymbol{\frac{(2r)^4}{2r}}$
# Explanation:
## Step1: Expand $(2r)^4$
$(2r)^4 = 2^4r^4=16r^4$
## Step2: Divide by $2r$
$\frac{16r^4}{2r}=\frac{16}{2}	imes\frac{r^4}{r}=8r^{4 - 1}=8r^3$
# Answer: $8r^3$

### Problem 12: $\boldsymbol{\frac{(3x^2)^3}{(3x^3)^2}}$
# Explanation:
## Step1: Expand numerator and denominator
Numerator: $(3x^2)^3=3^3(x^2)^3 = 27x^6$; Denominator: $(3x^3)^2=3^2(x^3)^2 = 9x^6$
## Step2: Divide numerator by denominator
$\frac{27x^6}{9x^6}=\frac{27}{9}	imes\frac{x^6}{x^6}=3	imes1 = 3$
# Answer: $3$

### Problem 13: $\boldsymbol{\frac{(2ab)^2}{2ab^2}}$
# Explanation:
## Step1: Expand $(2ab)^2$
$(2ab)^2=4a^2b^2$
## Step2: Divide by $2ab^2$
$\frac{4a^2b^2}{2ab^2}=\frac{4}{2}	imes\frac{a^2}{a}	imes\frac{b^2}{b^2}=2a^{2 - 1}	imes1 = 2a$
# Answer: $2a$

### Problem 14: $\boldsymbol{\frac{(-x)^9}{(-x)^6}}$
# Explanation:
## Step1: Use exponent rule $a^m\div a^n=a^{m - n}$
Here, $a=-x$, $m = 9$, $n = 6$. So, $(-x)^{9 - 6}=(-x)^3=-x^3$
# Answer: $-x^3$

### Problem 15: $\boldsymbol{\frac{(-xy)^9}{xy^9}}$
# Explanation:
## Step1: Expand $(-xy)^9$
$(-xy)^9=(-1)^9x^9y^9=-x^9y^9$
## Step2: Divide by $xy^9$
$\frac{-x^9y^9}{xy^9}=- \frac{x^9}{x}	imes\frac{y^9}{y^9}=-x^{9 - 1}	imes1=-x^8$
# Answer: $-x^8$

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QUESTION IMAGE

Question

Explanation:

Problem 1: $\boldsymbol{\frac{42}{63}}$

Step1: Find GCD of 42 and 63

Step2: Divide numerator and denominator by 21

Step1: Use exponent rule $a^m\div a^n = a^{m - n}$

Step2: Simplify $10^{-2}$

Step1: Divide coefficients and variables separately

Step2: Simplify

Answer:

Problem 2: $\boldsymbol{\frac{10^4}{10^6}}$