Question

name____ date__ period____
properties of exponents quotient rule
directions: circle the correct answer for each question below.

question	a	b	c	d
2. simplify the quotient below: $\frac{4^{18}}{4^{16}}$	$1^{2}$ (orange)	$4^{2}$ (yellow)	$16^{2}$ (red)	$4^{34}$ (green)
3. simplify the quotient below: $\frac{b^{11}}{b^{3}}$	$b^{14}$ (blue)	$1^{14}$ (orange)	$b^{7}$ (red)	$b^{8}$ (green)
4. simplify the quotient below: $\frac{x^{6}}{x^{2}}$	$x^{2}$ (purple)	$x^{4}$ (orange)	$\frac{1}{x^{8}}$ (white)	$x^{8}$ (yellow)
5. simplify the quotient: $\frac{6y^{2}}{3y}$	$2y^{2}$ (pink)	$2y$ (green)	$3y^{2}$ (orange)	$\frac{2}{y}$ (purple)
6. simplify the quotient: $\frac{-27a^{5}}{18a^{3}}$	$\frac{-3a^{2}}{2}$ (purple)	$\frac{-3}{4a^{2}}$ (blue)	$\frac{-3a^{8}}{2}$ (green)	$\frac{-3a^{2}}{4}$ (red)
7. simplify the quotient: $\frac{x^{8}y^{4}}{x^{7}y^{2}}$	$\frac{x}{y^{2}}$ (white)	$xy^{2}$ (green)	$x^{2}y^{2}$ (blue)	$x^{15}y^{6}$ (pink)
8. simplify the quotient: $\frac{14a^{13}b^{9}}{42a^{9}b^{2}}$	$\frac{a^{4}b^{7}}{3}$ (red)	$-3a^{22}b^{11}$ (white)	$\frac{-3a^{22}}{b^{7}}$ (yellow)	$\frac{b^{7}}{3a^{4}}$ (purple)
9. simplify the quotient: $\frac{a^{7}b^{7}c^{6}}{a^{3}b^{4}c^{5}}$	$a^{10}b^{11}c^{11}$ (green)	$\frac{a^{4}b^{3}}{c}$ (pink)	$a^{4}b^{3}c$ (orange)	$abc$ (red)
10. find the value of y. $\frac{2^{y}}{2^{7}} = 2^{8}$	$y = 7$ (blue)	$y = 15$ (purple)	$y = 8$ (yellow)	$y = 1$ (white)
11. find the value of a. $\frac{x^{5}}{x^{a}} = x^{2}$	$a = 5$ (purple)	$a = 2$ (orange)	$a = 3$ (yellow)	$a = 7$ (blue)
12. find the value of x: $\frac{a^{3}b^{5}c^{x}}{ab^{2}c^{2}} = a^{2}b^{3}c^{2}$	$x = 4$ (tan)	$x = 2$ (green)	$x = 0$ (red)	$x = 1$ (yellow)
13. find the quotient: $\frac{2x^{5}y^{9}z^{13}}{x^{2}y^{3}z^{4}}$	$2x^{3}y^{6}z^{9}$ (red)	$\frac{2}{x^{3}y^{6}z^{9}}$ (yellow)	$x^{3}y^{6}z^{9}$ (green)	$2x^{3}y^{6}$ (orange)
14. simplify the quotient: $\frac{-30r^{8}}{10r^{6}}$	$\frac{-3}{r^{2}}$ (white)	$-30r^{2}$ (green)	$\frac{-30}{r^{2}}$ (purple)	$-3r^{2}$ (red)

Explanation:

Step1: Recall quotient - rule of exponents

The quotient - rule states that $\frac{a^m}{a^n}=a^{m - n}$ for $a
eq0$, where $m$ and $n$ are real numbers.

Step2: Solve problem 1

For $\frac{9^8}{9^5}$, using the quotient - rule, we have $9^{8 - 5}=9^3$.

Step3: Solve problem 2

For $\frac{4^{18}}{4^{16}}$, by the quotient - rule, $4^{18-16}=4^2$.

Step4: Solve problem 3

For $\frac{b^{11}}{b^3}$, applying the quotient - rule, $b^{11 - 3}=b^8$.

Step5: Solve problem 4

For $\frac{x^6}{x^2}$, using the quotient - rule, $x^{6 - 2}=x^4$.

Step6: Solve problem 5

First, simplify $\frac{6y^2}{3y}=\frac{6}{3}\times\frac{y^2}{y}=2y$.

Step7: Solve problem 6

For $\frac{-27a^5}{18a^3}$, first simplify the coefficient $\frac{-27}{18}=-\frac{3}{2}$, and for the variable part $\frac{a^5}{a^3}=a^{5 - 3}=a^2$. So the result is $-\frac{3a^2}{2}$.

Step8: Solve problem 7

For $\frac{x^8y^4}{x^7y^2}$, for the $x$ - part $\frac{x^8}{x^7}=x^{8 - 7}=x$, and for the $y$ - part $\frac{y^4}{y^2}=y^{4 - 2}=y^2$. So the result is $xy^2$.

Step9: Solve problem 8

For $\frac{14a^{13}b^9}{42a^9b^2}$, simplify the coefficient $\frac{14}{42}=\frac{1}{3}$, for the $a$ - part $\frac{a^{13}}{a^9}=a^{13 - 9}=a^4$, and for the $b$ - part $\frac{b^9}{b^2}=b^{9 - 2}=b^7$. So the result is $\frac{a^4b^7}{3}$.

Step10: Solve problem 9

For $\frac{a^7b^7c^6}{a^3b^4c^5}$, for the $a$ - part $\frac{a^7}{a^3}=a^{7 - 3}=a^4$, for the $b$ - part $\frac{b^7}{b^4}=b^{7 - 4}=b^3$, and for the $c$ - part $\frac{c^6}{c^5}=c^{6 - 5}=c$. So the result is $a^4b^3c$.

Step11: Solve problem 10

Given $\frac{2^y}{2^7}=2^8$, using the quotient - rule $\frac{2^y}{2^7}=2^{y - 7}$, so $y-7 = 8$, then $y=15$.

Step12: Solve problem 11

Given $\frac{x^5}{x^a}=x^2$, using the quotient - rule $\frac{x^5}{x^a}=x^{5 - a}$, so $5 - a=2$, then $a = 3$.

Step13: Solve problem 12

For $\frac{a^3b^5c^x}{ab^2c^2}=a^2b^3c^2$, for the $a$ - part $\frac{a^3}{a}=a^{3 - 1}=a^2$, for the $b$ - part $\frac{b^5}{b^2}=b^{5 - 2}=b^3$, for the $c$ - part $\frac{c^x}{c^2}=c^{x - 2}$, so $x-2 = 2$, then $x = 4$.

Step14: Solve problem 13

For $\frac{2x^5y^9z^{13}}{x^2y^3z^4}$, for the coefficient it remains $2$, for the $x$ - part $\frac{x^5}{x^2}=x^{5 - 2}=x^3$, for the $y$ - part $\frac{y^9}{y^3}=y^{9 - 3}=y^6$, and for the $z$ - part $\frac{z^{13}}{z^4}=z^{13 - 4}=z^9$. So the result is $2x^3y^6z^9$.

Step15: Solve problem 14

For $\frac{-30r^8}{10r^6}$, simplify the coefficient $\frac{-30}{10}=-3$, and for the variable part $\frac{r^8}{r^6}=r^{8 - 6}=r^2$. So the result is $-3r^2$.

Answer:

1. C. $9^3$
2. B. $4^2$
3. D. $b^8$
4. B. $x^4$
5. B. $2y$
6. A. $-\frac{3a^2}{2}$
7. B. $xy^2$
8. A. $\frac{a^4b^7}{3}$
9. C. $a^4b^3c$
10. B. $y = 15$
11. C. $a = 3$
12. A. $x = 4$
13. A. $2x^3y^6z^9$
14. D. $-3r^2$