To solve these problems, we use the
Quotient Rule for Exponents.
The Rule:
When dividing terms with the same base, you
subtract the bottom exponent from the top exponent.
$$ \frac{a^m}{a^n} = a^{m-n} $$
Let's work through each problem step-by-step:
1.
$\frac{2^5}{2^2}$
* Subtract exponents: $5 - 2 = 3$
* Result: $2^3$ (or 8)
2.
$\frac{5^4}{5^2}$
* Subtract exponents: $4 - 2 = 2$
* Result: $5^2$ (or 25)
3.
$\frac{4^8}{4^5}$
* Subtract exponents: $8 - 5 = 3$
* Result: $4^3$ (or 64)
4.
$\frac{8^6}{8^2}$
* Subtract exponents: $6 - 2 = 4$
* Result: $8^4$ (or 4096)
5.
$\frac{7^2}{7^3}$
* Subtract exponents: $2 - 3 = -1$
* Result: $7^{-1}$ (or $\frac{1}{7}$)
6.
$\frac{x^7}{x^3}$
* Subtract exponents: $7 - 3 = 4$
* Result: $x^4$
7.
$\frac{x^2}{x^{10}}$
* Subtract exponents: $2 - 10 = -8$
* Result: $x^{-8}$ (or $\frac{1}{x^8}$)
8.
$\frac{q^3}{q^{-5}}$
* Subtract exponents: $3 - (-5) = 3 + 5 = 8$
* Result: $q^8$
9.
$\frac{9^2}{9^4}$
* Subtract exponents: $2 - 4 = -2$
* Result: $9^{-2}$ (or $\frac{1}{9^2} = \frac{1}{81}$)
10.
$\frac{6^3}{6^6}$
* Subtract exponents: $3 - 6 = -3$
* Result: $6^{-3}$ (or $\frac{1}{6^3} = \frac{1}{216}$)
11.
$\frac{25^2}{25}$
* Note: $25$ is the same as $25^1$.
* Subtract exponents: $2 - 1 = 1$
* Result: $25^1$ or just $25$
12.
$\frac{x^{36}}{x^{27}}$
* Subtract exponents: $36 - 27 = 9$
* Result: $x^9$
13.
$\frac{4^{16}}{4^{12}}$
* Subtract exponents: $16 - 12 = 4$
* Result: $4^4$ (or 256)
14.
$\frac{2^3 \times 2^6}{2^{18}}$
* First, simplify the numerator using the Product Rule (add exponents): $2^3 \times 2^6 = 2^{3+6} = 2^9$.
* Now divide: $\frac{2^9}{2^{18}}$.
* Subtract exponents: $9 - 18 = -9$.
* Result: $2^{-9}$ (or $\frac{1}{2^9} = \frac{1}{512}$)
Final Answer:
1. $2^3$ (or 8)
2. $5^2$ (or 25)
3. $4^3$ (or 64)
4. $8^4$ (or 4096)
5. $7^{-1}$ (or $\frac{1}{7}$)
6. $x^4$
7. $x^{-8}$ (or $\frac{1}{x^8}$)
8. $q^8$
9. $9^{-2}$ (or $\frac{1}{81}$)
10. $6^{-3}$ (or $\frac{1}{216}$)
11. $25$
12. $x^9$
13. $4^4$ (or 256)
14. $2^{-9}$ (or $\frac{1}{512}$)
Parent Tip: Review the logic above to help your child master the concept of dividing exponents worksheet.
