Practice Worksheet for Law of Exponents Use the laws of ... - wveis - Free Printable

Let's solve each problem on the Practice Worksheet for Law of Exponents step by step, using the laws of exponents. We'll simplify each expression and write the answer with positive exponents.

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🔷 Laws of Exponents Recap:

1. $ a^m \cdot a^n = a^{m+n} $
2. $ \frac{a^m}{a^n} = a^{m-n} $
3. $ (a^m)^n = a^{m \cdot n} $
4. $ (ab)^n = a^n b^n $
5. $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $
6. $ a^{-n} = \frac{1}{a^n} $
7. $ \frac{1}{a^{-n}} = a^n $

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Now let’s go through each problem:

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1. $ 3^1 \cdot 3^4 $

Use: $ a^m \cdot a^n = a^{m+n} $

$$
3^{1+4} = 3^5
$$

✔ Answer: $ 3^5 $

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2. $ x^4 \cdot x^2 $

$$
x^{4+2} = x^6
$$

✔ Answer: $ x^6 $

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3. $ 3x^2 \cdot 4x^2 $

Multiply coefficients and add exponents:

$$
(3 \cdot 4)(x^2 \cdot x^2) = 12x^{2+2} = 12x^4
$$

✔ Answer: $ 12x^4 $

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4. $ x^2 y^4 \cdot x^3 y^{10} $

Group like terms:

$$
x^{2+3} y^{4+10} = x^5 y^{14}
$$

✔ Answer: $ x^5 y^{14} $

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5. $ (5^2)^3 $

Use: $ (a^m)^n = a^{m \cdot n} $

$$
5^{2 \cdot 3} = 5^6
$$

✔ Answer: $ 5^6 $

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6. $ (x^4)^3 $

$$
x^{4 \cdot 3} = x^{12}
$$

✔ Answer: $ x^{12} $

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7. $ (2x)^4 $

Use: $ (ab)^n = a^n b^n $

$$
2^4 \cdot x^4 = 16x^4
$$

✔ Answer: $ 16x^4 $

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8. $ (x^4 y^3)^3 $

$$
x^{4 \cdot 3} y^{3 \cdot 3} = x^{12} y^9
$$

✔ Answer: $ x^{12} y^9 $

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9. $ (3x^4 y z^2)^2 $

$$
3^2 \cdot x^{4 \cdot 2} \cdot y^2 \cdot z^{2 \cdot 2} = 9x^8 y^2 z^4
$$

✔ Answer: $ 9x^8 y^2 z^4 $

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10. $ (2x^3)^3 (-3x^2 y^3)^2 $

Break into parts:

- $ (2x^3)^3 = 2^3 x^{9} = 8x^9 $
- $ (-3x^2 y^3)^2 = (-3)^2 x^{4} y^{6} = 9x^4 y^6 $

Now multiply:

$$
8x^9 \cdot 9x^4 y^6 = 72x^{9+4} y^6 = 72x^{13} y^6
$$

✔ Answer: $ 72x^{13} y^6 $

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11. $ 3^{-4} $

Negative exponent → move to denominator:

$$
\frac{1}{3^4} = \frac{1}{81}
$$

✔ Answer: $ \frac{1}{81} $

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12. $ \frac{8^5}{8^3} $

Use: $ \frac{a^m}{a^n} = a^{m-n} $

$$
8^{5-3} = 8^2 = 64
$$

✔ Answer: $ 64 $

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13. $ 27^2 $

Note: $ 27 = 3^3 $, so:

$$
(3^3)^2 = 3^{6} = 729
$$

Alternatively, $ 27^2 = 27 \cdot 27 = 729 $

✔ Answer: $ 729 $

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14. $ \frac{12x^5}{4x^2} $

Simplify coefficients and subtract exponents:

$$
\frac{12}{4} \cdot x^{5-2} = 3x^3
$$

✔ Answer: $ 3x^3 $

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15. $ \frac{2x^2 y^3}{16x^2 y} $

Simplify:

- Coefficients: $ \frac{2}{16} = \frac{1}{8} $
- $ x^{2-2} = x^0 = 1 $
- $ y^{3-1} = y^2 $

So:

$$
\frac{1}{8} y^2 = \frac{y^2}{8}
$$

✔ Answer: $ \frac{y^2}{8} $

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16. $ \frac{3^5}{3^7} $

$$
3^{5-7} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}
$$

✔ Answer: $ \frac{1}{9} $

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17. $ \frac{4x^2 y^5}{20x^2 y^3} $

Simplify:

- Coefficients: $ \frac{4}{20} = \frac{1}{5} $
- $ x^{2-2} = x^0 = 1 $
- $ y^{5-3} = y^2 $

$$
\frac{1}{5} y^2 = \frac{y^2}{5}
$$

✔ Answer: $ \frac{y^2}{5} $

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18. $ \frac{12xy^3}{3x^3 y^2} $

- Coefficients: $ \frac{12}{3} = 4 $
- $ x^{1-3} = x^{-2} $
- $ y^{3-2} = y^1 $

So: $ 4x^{-2} y $

Convert negative exponent:

$$
\frac{4y}{x^2}
$$

✔ Answer: $ \frac{4y}{x^2} $

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19. $ \left(\frac{2x^4}{3x}\right)^3 $

Simplify inside first:

$$
\frac{2x^4}{3x} = \frac{2}{3} x^{4-1} = \frac{2}{3} x^3
$$

Now cube it:

$$
\left(\frac{2}{3} x^3\right)^3 = \left(\frac{2}{3}\right)^3 \cdot (x^3)^3 = \frac{8}{27} x^9
$$

✔ Answer: $ \frac{8}{27} x^9 $

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20. $ \frac{18x^{-3} y^4}{12x^{-1} y^{-3}} $

Simplify coefficients and use exponent rules:

- $ \frac{18}{12} = \frac{3}{2} $
- $ x^{-3 - (-1)} = x^{-3 + 1} = x^{-2} $
- $ y^{4 - (-3)} = y^{4+3} = y^7 $

So: $ \frac{3}{2} x^{-2} y^7 $

Convert negative exponent:

$$
\frac{3y^7}{2x^2}
$$

✔ Answer: $ \frac{3y^7}{2x^2} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ 3^5 $ |
| 2 | $ x^6 $ |
| 3 | $ 12x^4 $ |
| 4 | $ x^5 y^{14} $ |
| 5 | $ 5^6 $ |
| 6 | $ x^{12} $ |
| 7 | $ 16x^4 $ |
| 8 | $ x^{12} y^9 $ |
| 9 | $ 9x^8 y^2 z^4 $ |
| 10 | $ 72x^{13} y^6 $ |
| 11 | $ \frac{1}{81} $ |
| 12 | $ 64 $ |
| 13 | $ 729 $ |
| 14 | $ 3x^3 $ |
| 15 | $ \frac{y^2}{8} $ |
| 16 | $ \frac{1}{9} $ |
| 17 | $ \frac{y^2}{5} $ |
| 18 | $ \frac{4y}{x^2} $ |
| 19 | $ \frac{8}{27} x^9 $ |
| 20 | $ \frac{3y^7}{2x^2} $ |

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