simplifying fraction exponent expressions (division) Math ... - Free Printable

Of course! Let's solve each of the 15 problems on this worksheet step by step. The key rules we'll use are:

- When multiplying powers with the same base: $ a^m \cdot a^n = a^{m+n} $
- When dividing powers with the same base: $ \frac{a^m}{a^n} = a^{m-n} $
- Any number to the power of 0 is 1: $ a^0 = 1 $

We’ll simplify each expression by combining exponents in the numerator and denominator separately, then subtracting the denominator’s exponent from the numerator’s.

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Problem 1

$$
\frac{\left(\frac{2}{5}\right)^2 \cdot \left(\frac{2}{5}\right)^2 \cdot \left(\frac{2}{5}\right)^4 \cdot \left(\frac{2}{5}\right)^{-5}}{\left(\frac{2}{5}\right)^{-2} \cdot \left(\frac{2}{5}\right)^{-4}}
$$

Numerator: $ 2 + 2 + 4 + (-5) = 3 $

Denominator: $ -2 + (-4) = -6 $

Result: $ \left(\frac{2}{5}\right)^{3 - (-6)} = \left(\frac{2}{5}\right)^9 $

✔ Answer: $ \left(\frac{2}{5}\right)^9 $

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Problem 2

$$
\frac{\left(\frac{1}{7}\right)^{-4} \cdot \left(\frac{1}{7}\right)^9 \cdot \left(\frac{1}{7}\right)^{-4} \cdot \left(\frac{1}{7}\right)^{-2}}{\left(\frac{1}{7}\right)^{-10} \cdot \left(\frac{1}{7}\right)^{-4}}
$$

Numerator: $ -4 + 9 + (-4) + (-2) = -1 $

Denominator: $ -10 + (-4) = -14 $

Result: $ \left(\frac{1}{7}\right)^{-1 - (-14)} = \left(\frac{1}{7}\right)^{13} $

✔ Answer: $ \left(\frac{1}{7}\right)^{13} $

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Problem 3

$$
\frac{\left(\frac{1}{2}\right)^{10} \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right)^6}{\left(\frac{1}{2}\right)^{-2}}
$$

Numerator: $ 10 + 1 + 6 = 17 $

Denominator: $ -2 $

Result: $ \left(\frac{1}{2}\right)^{17 - (-2)} = \left(\frac{1}{2}\right)^{19} $

✔ Answer: $ \left(\frac{1}{2}\right)^{19} $

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Problem 4

$$
\frac{\left(\frac{2}{7}\right)^6 \cdot \left(\frac{2}{7}\right)^{-5} \cdot \left(\frac{2}{7}\right)^{-7} \cdot \left(\frac{2}{7}\right)^3}{\left(\frac{2}{7}\right)^{-4} \cdot \left(\frac{2}{7}\right)^{-2}}
$$

Numerator: $ 6 + (-5) + (-7) + 3 = -3 $

Denominator: $ -4 + (-2) = -6 $

Result: $ \left(\frac{2}{7}\right)^{-3 - (-6)} = \left(\frac{2}{7}\right)^3 $

✔ Answer: $ \left(\frac{2}{7}\right)^3 $

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Problem 5

$$
\frac{\left(\frac{4}{5}\right)^2 \cdot \left(\frac{4}{5}\right)^{-2} \cdot \left(\frac{4}{5}\right)^9 \cdot \left(\frac{4}{5}\right)^9}{\left(\frac{4}{5}\right)^{-10} \cdot \left(\frac{4}{5}\right)^4}
$$

Numerator: $ 2 + (-2) + 9 + 9 = 18 $

Denominator: $ -10 + 4 = -6 $

Result: $ \left(\frac{4}{5}\right)^{18 - (-6)} = \left(\frac{4}{5}\right)^{24} $

✔ Answer: $ \left(\frac{4}{5}\right)^{24} $

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Problem 6

$$
\frac{\left(\frac{4}{9}\right)^{10} \cdot \left(\frac{4}{9}\right)^{11} \cdot \left(\frac{4}{9}\right)^{-5} \cdot \left(\frac{4}{9}\right)^{-4}}{\left(\frac{4}{9}\right)^{-1} \cdot \left(\frac{4}{9}\right)^{-8}}
$$

Numerator: $ 10 + 11 + (-5) + (-4) = 12 $

Denominator: $ -1 + (-8) = -9 $

Result: $ \left(\frac{4}{9}\right)^{12 - (-9)} = \left(\frac{4}{9}\right)^{21} $

✔ Answer: $ \left(\frac{4}{9}\right)^{21} $

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Problem 7

$$
\frac{\left(\frac{1}{2}\right)^{-3} \cdot \left(\frac{1}{2}\right)^4 \cdot \left(\frac{1}{2}\right)^4}{\left(\frac{1}{2}\right)^{10}}
$$

Numerator: $ -3 + 4 + 4 = 5 $

Denominator: $ 10 $

Result: $ \left(\frac{1}{2}\right)^{5 - 10} = \left(\frac{1}{2}\right)^{-5} = 2^5 = 32 $

✔ Answer: $ 32 $ *(or $ \left(\frac{1}{2}\right)^{-5} $, but usually simplified to positive exponent)*

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Problem 8

$$
\left(\frac{4}{9}\right)^5 \cdot \left(\frac{4}{9}\right)^{-2} \cdot \left(\frac{4}{9}\right)^{-4}
$$

This one has no denominator — just multiply.

Exponent: $ 5 + (-2) + (-4) = -1 $

✔ Answer: $ \left(\frac{4}{9}\right)^{-1} = \frac{9}{4} $

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Problem 9

$$
\frac{\left(\frac{1}{3}\right)^3 \cdot \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right)^3}{\left(\frac{1}{3}\right)^{-3}}
$$

Numerator: $ 3 + 1 + 3 = 7 $

Denominator: $ -3 $

Result: $ \left(\frac{1}{3}\right)^{7 - (-3)} = \left(\frac{1}{3}\right)^{10} $

✔ Answer: $ \left(\frac{1}{3}\right)^{10} $

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Problem 10

$$
\left(\frac{2}{9}\right)^2 \cdot \left(\frac{2}{9}\right)^8 \cdot \left(\frac{2}{9}\right)^{-1}
$$

Exponent: $ 2 + 8 + (-1) = 9 $

✔ Answer: $ \left(\frac{2}{9}\right)^9 $

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Problem 11

$$
\frac{\left(\frac{2}{9}\right)^{-10} \cdot \left(\frac{2}{9}\right)^2 \cdot \left(\frac{2}{9}\right)^{-5} \cdot \left(\frac{2}{9}\right)^4}{\left(\frac{2}{9}\right)^6 \cdot \left(\frac{2}{9}\right)^{-10}}
$$

Numerator: $ -10 + 2 + (-5) + 4 = -9 $

Denominator: $ 6 + (-10) = -4 $

Result: $ \left(\frac{2}{9}\right)^{-9 - (-4)} = \left(\frac{2}{9}\right)^{-5} = \left(\frac{9}{2}\right)^5 $

✔ Answer: $ \left(\frac{9}{2}\right)^5 $

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Problem 12

$$
\left(\frac{1}{5}\right)^{-3} \cdot \left(\frac{1}{5}\right)^5 \cdot \left(\frac{1}{5}\right)^{-3}
$$

Exponent: $ -3 + 5 + (-3) = -1 $

✔ Answer: $ \left(\frac{1}{5}\right)^{-1} = 5 $

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Problem 13

$$
\frac{\left(\frac{1}{3}\right)^{-7} \cdot \left(\frac{1}{3}\right)^{-10} \cdot \left(\frac{1}{3}\right)^{-9}}{\left(\frac{1}{3}\right)^{-10}}
$$

Numerator: $ -7 + (-10) + (-9) = -26 $

Denominator: $ -10 $

Result: $ \left(\frac{1}{3}\right)^{-26 - (-10)} = \left(\frac{1}{3}\right)^{-16} = 3^{16} $

✔ Answer: $ 3^{16} $

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Problem 14

$$
\left(\frac{3}{8}\right)^6 \cdot \left(\frac{3}{8}\right)^{-1} \cdot \left(\frac{3}{8}\right)
$$

Exponent: $ 6 + (-1) + 1 = 6 $

✔ Answer: $ \left(\frac{3}{8}\right)^6 $

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Problem 15

$$
\frac{\left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right)^7 \cdot \left(\frac{1}{3}\right)^5}{\left(\frac{1}{3}\right)^9}
$$

Numerator: $ 1 + 7 + 5 = 13 $

Denominator: $ 9 $

Result: $ \left(\frac{1}{3}\right)^{13 - 9} = \left(\frac{1}{3}\right)^4 $

✔ Answer: $ \left(\frac{1}{3}\right)^4 $

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## ✔ Final Answers (Boxed):

1. $ \boxed{\left(\frac{2}{5}\right)^9} $
2. $ \boxed{\left(\frac{1}{7}\right)^{13}} $
3. $ \boxed{\left(\frac{1}{2}\right)^{19}} $
4. $ \boxed{\left(\frac{2}{7}\right)^3} $
5. $ \boxed{\left(\frac{4}{5}\right)^{24}} $
6. $ \boxed{\left(\frac{4}{9}\right)^{21}} $
7. $ \boxed{32} $
8. $ \boxed{\frac{9}{4}} $
9. $ \boxed{\left(\frac{1}{3}\right)^{10}} $
10. $ \boxed{\left(\frac{2}{9}\right)^9} $
11. $ \boxed{\left(\frac{9}{2}\right)^5} $
12. $ \boxed{5} $
13. $ \boxed{3^{16}} $
14. $ \boxed{\left(\frac{3}{8}\right)^6} $
15. $ \boxed{\left(\frac{1}{3}\right)^4} $

Let me know if you’d like these written as decimals or expanded forms too!