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 $

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

✔ Answer: $ x^{12} y^9 $

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

$$
3^2 \cdot (x^4)^2 \cdot y^2 \cdot (z^2)^2 = 9x^8 y^2 z^4
$$

✔ 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 \cdot (x^3)^3 = 8x^9 $
- $ (-3x^2 y^3)^2 = (-3)^2 \cdot (x^2)^2 \cdot (y^3)^2 = 9x^4 y^6 $

Now multiply:

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

✔ Answer: $ 72x^{13} y^6 $

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11. $ 3^{-4} $

Use: $ a^{-n} = \frac{1}{a^n} $

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

✔ 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:

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

Alternatively, compute directly: $ 27^2 = 729 $

But since it says "simplify with positive exponents", we can leave as $ 27^2 $ or $ 729 $. But likely they want simplified form.

So:

$$
27^2 = 729
$$

✔ Answer: $ 729 $

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

Simplify coefficients and variables:

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

$$
3x^3
$$

✔ Answer: $ 3x^3 $

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

Simplify numerator and denominator:

- $ \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:

- $ \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} $

Simplify:

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

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

Now write with positive exponents:

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

✔ Answer: $ \frac{4y}{x^2} $

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

First simplify inside the parentheses:

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

Now raise to power 3:

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

✔ Answer: $ \frac{8}{27} x^9 $

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20. $ \frac{18x^{-3} y^4}{12x^{-1} y^{-2}} $

Simplify coefficients and variables:

- $ \frac{18}{12} = \frac{3}{2} $
- $ x^{-3 - (-1)} = x^{-3 + 1} = x^{-2} $
- $ y^{4 - (-2)} = y^{4 + 2} = y^6 $

So: $ \frac{3}{2} x^{-2} y^6 $

Now write with positive exponents:

$$
\frac{3y^6}{2x^2}
$$

✔ Answer: $ \frac{3y^6}{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^6}{2x^2} $ |

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Let me know if you'd like these formatted in a printable sheet!