Quotient Rule for Exponents

Show your work for each problem.

1.
Simplify $\frac{7^9}{7^3}$ .
1. a) $7^6$
2. b) $7^3$
3. c) $7^{12}$
4. d) $1^6$
Answer: $7^6$
$\frac{7^9}{7^3} = 7^{9-3} = 7^6$ . The exponents are subtracted, not divided ( $7^3$ ) or added ( $7^{12}$ ), and the base stays $7$ .
2.
$\frac{x^{11}}{x^4} = x^{\square}$ . What number goes in the box?
Answer: 7
$4$ of the $11$ factors of $x$ cancel, so $11 - 4 = 7$ stay: $x^7$ .
3.
Simplify $\frac{2^8}{2^5}$ and evaluate it.
Answer: 8
1. Subtract the exponents: $8 - 5$ 3
2. Evaluate $2^3$ 8
4.
Which expression equals $\frac{a^{10}}{a^2}$ (with $a \neq 0$ )?
1. a) $a^{12}$
2. b) $a^5$
3. c) $a^{20}$
4. d) $a^8$
Answer: $a^8$
$\frac{a^{10}}{a^2} = a^{10-2} = a^8$ . The choice $a^5$ comes from dividing the exponents, which is the most common mistake.
5.
Evaluate $\frac{10^9}{10^6}$ .
Answer: 1000
$\frac{10^9}{10^6} = 10^{9-6} = 10^3 = 1000$ .
6.
Simplify $\frac{y^7}{y^2}$ by cancelling factors.
Answer: y^5
1. How many factors of $y$ are on top? 7
2. How many factors cancel? 2
3. How many factors of $y$ are left? This is the new exponent. 5
7.
Ana simplified $\frac{x^{12}}{x^3}$ and got $x^4$ . What did she do wrong?
1. a)She divided the bases
2. b)She subtracted the exponents in the wrong order
3. c)She divided the exponents instead of subtracting them
4. d)Nothing, $x^4$ is correct
Answer: She divided the exponents instead of subtracting them
$12 \div 3 = 4$ : Ana divided the exponents. Correct is $\frac{x^{12}}{x^3} = x^{12-3} = x^9$ .
8.
$\frac{15x^8}{3x^5} = \square\, x^3$ . What number goes in the box?
Answer: 5
$15 \div 3 = 5$ and $x^{8-5} = x^3$ , so $\frac{15x^8}{3x^5} = 5x^3$ .
9.
Simplify $\frac{20m^9}{5m^4}$ .
Answer: 4m^5
1. Divide the coefficients: $20 \div 5$ 4
2. What is the exponent of $m$ in the result? 5
10.
Simplify $\frac{24a^7b^4}{8a^3b}$ .
1. a) $16a^4b^3$
2. b) $3a^4b^3$
3. c) $3a^{10}b^5$
4. d) $3a^4b^4$
Answer: $3a^4b^3$
$24 \div 8 = 3$ , $a^{7-3} = a^4$ and $b^{4-1} = b^3$ , so the result is $3a^4b^3$ .
11.
Evaluate $\frac{3^{10}}{3^7}$ .
Answer: 27
$\frac{3^{10}}{3^7} = 3^{10-7} = 3^3 = 27$ , without calculating $3^{10} = 59049$ .
12.
Simplify $\frac{18p^6q^5}{6p^2q^3}$ .
Answer: 3p^4q^2
1. Divide the coefficients: $18 \div 6$ 3
2. What is the exponent of $p$ in the result? 4
3. What is the exponent of $q$ in the result? 2
13.
Simplify $\frac{4^3 \cdot 4^5}{4^6}$ and evaluate it.
Answer: 16
1. Product rule on top: what is the exponent of $4^3 \cdot 4^5$ ? 8
2. Quotient rule: what is the exponent of $\frac{4^8}{4^6}$ ? 2
3. Evaluate $4^2$ 16
14.
Which expression is NOT equal to $x^6$ (with $x \neq 0$ )?
1. a) $\frac{x^2 \cdot x^7}{x^3}$
2. b) $\frac{x^9}{x^3}$
3. c) $\frac{x^{12}}{x^2}$
4. d) $\frac{x^{10}}{x^4}$
Answer: $\frac{x^{12}}{x^2}$
$\frac{x^9}{x^3} = x^6$ , $\frac{x^2 \cdot x^7}{x^3} = x^{9-3} = x^6$ and $\frac{x^{10}}{x^4} = x^6$ , but $\frac{x^{12}}{x^2} = x^{10}$ . (Dividing $12$ by $2$ would give $6$ , which is the trap.)
15.
$\frac{x^{\square}}{x^4} = x^9$ . What number goes in the box?
Answer: 13
The rule gives $\square - 4 = 9$ , so $\square = 13$ . Check: $\frac{x^{13}}{x^4} = x^{13-4} = x^9$ .
16.
Simplify $\frac{6x^5 \cdot 3x^4}{9x^7}$ .
Answer: 2x^2
1. Multiply the coefficients on top: $6 \cdot 3$ 18
2. What is the exponent of $x^5 \cdot x^4$ ? 9
3. Divide the coefficients: $18 \div 9$ 2
4. What is the exponent of $x$ in the result? 2
17.
Simplify $\frac{5a^3 \cdot 4a^6}{10a^2 \cdot a^4}$ .
Answer: 2a^3
1. Coefficient of the numerator: $5 \cdot 4$ 20
2. Exponent of $a$ in the numerator 9
3. Exponent of $a$ in the denominator ( $a^2 \cdot a^4$ ) 6
4. Coefficient of the result: $20 \div 10$ 2
5. Exponent of $a$ in the result 3
18.
A hard drive holds $2^{40}$ bytes. One film takes $2^{32}$ bytes. How many films fit on the drive?
Answer: 256
1. What is the exponent of $\frac{2^{40}}{2^{32}}$ ? 8
2. Evaluate $2^8$ 256