Algebra 1 Worksheets | Exponents Worksheets - Free Printable

Let's solve each problem step by step using the laws of exponents. The key rules we'll use are:

1. $ a^m \cdot a^n = a^{m+n} $
2. $ (a^m)^n = a^{m \cdot n} $
3. $ (ab)^n = a^n b^n $
4. $ a^{-n} = \frac{1}{a^n} $
5. For expressions with coefficients, multiply the coefficients and add exponents for like bases.

We’ll simplify each expression so that the final answer contains only positive exponents.

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1) $ 9^4 \cdot 9^3 $

Same base: $ 9 $

Add exponents:
$ 9^{4+3} = 9^7 $

✔ Answer: $ \boxed{9^7} $

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

Same base: $ \frac{1}{w} $

Add exponents:
$ \left(\frac{1}{w}\right)^{6+2} = \left(\frac{1}{w}\right)^8 = \frac{1}{w^8} $

✔ Answer: $ \boxed{\frac{1}{w^8}} $

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3) $ r^3 s^2 \cdot 8r^4 s^4 \cdot 4rs^6 $

Group coefficients and variables:

- Coefficients: $ 1 \cdot 8 \cdot 4 = 32 $
- $ r^3 \cdot r^4 \cdot r = r^{3+4+1} = r^8 $
- $ s^2 \cdot s^4 \cdot s^6 = s^{2+4+6} = s^{12} $

✔ Answer: $ \boxed{32r^8 s^{12}} $

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4) $ 7y \cdot 2y^2 $

Coefficients: $ 7 \cdot 2 = 14 $

Variables: $ y \cdot y^2 = y^{1+2} = y^3 $

✔ Answer: $ \boxed{14y^3} $

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5) $ \left(\frac{1}{5}\right)^4 \cdot \left(\frac{1}{5}\right)^3 \cdot \left(\frac{1}{5}\right)^2 $

Same base: $ \frac{1}{5} $

Add exponents:
$ \left(\frac{1}{5}\right)^{4+3+2} = \left(\frac{1}{5}\right)^9 = \frac{1}{5^9} $

✔ Answer: $ \boxed{\frac{1}{5^9}} $

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6) $ 5a^3 n^6 \cdot 4a n^4 $

Coefficients: $ 5 \cdot 4 = 20 $

- $ a^3 \cdot a = a^{3+1} = a^4 $
- $ n^6 \cdot n^4 = n^{6+4} = n^{10} $

✔ Answer: $ \boxed{20a^4 n^{10}} $

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7) $ 8nr^4 \cdot 9n^6 r^3 $

Coefficients: $ 8 \cdot 9 = 72 $

- $ n \cdot n^6 = n^{1+6} = n^7 $
- $ r^4 \cdot r^3 = r^{4+3} = r^7 $

✔ Answer: $ \boxed{72n^7 r^7} $

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8) $ \left(\frac{1}{k}\right)^3 \cdot \left(\frac{1}{k}\right)^4 \cdot \left(\frac{1}{k}\right)^6 $

Same base: $ \frac{1}{k} $

Add exponents:
$ \left(\frac{1}{k}\right)^{3+4+6} = \left(\frac{1}{k}\right)^{13} = \frac{1}{k^{13}} $

✔ Answer: $ \boxed{\frac{1}{k^{13}}} $

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9) $ 5z^5 \cdot 6z^3 \cdot 3z^6 $

Coefficients: $ 5 \cdot 6 \cdot 3 = 90 $

Exponents of $ z $: $ z^{5+3+6} = z^{14} $

✔ Answer: $ \boxed{90z^{14}} $

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10) $ 9 \cdot 9^8 $

Same base: $ 9 $

$ 9^1 \cdot 9^8 = 9^{1+8} = 9^9 $

✔ Answer: $ \boxed{9^9} $

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11) $ 9b \cdot 6b^3 $

Coefficients: $ 9 \cdot 6 = 54 $

$ b \cdot b^3 = b^{1+3} = b^4 $

✔ Answer: $ \boxed{54b^4} $

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12) $ 9s^4 \cdot 7s^2 $

Coefficients: $ 9 \cdot 7 = 63 $

$ s^4 \cdot s^2 = s^{4+2} = s^6 $

✔ Answer: $ \boxed{63s^6} $

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13) $ 2a^2 c^6 \cdot 8a^6 c^2 $

Coefficients: $ 2 \cdot 8 = 16 $

- $ a^2 \cdot a^6 = a^{2+6} = a^8 $
- $ c^6 \cdot c^2 = c^{6+2} = c^8 $

✔ Answer: $ \boxed{16a^8 c^8} $

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14) $ s^2 \cdot s^6 $

Same base: $ s $

$ s^{2+6} = s^8 $

✔ Answer: $ \boxed{s^8} $

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## ✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ 9^7 $ |
| 2 | $ \frac{1}{w^8} $ |
| 3 | $ 32r^8 s^{12} $ |
| 4 | $ 14y^3 $ |
| 5 | $ \frac{1}{5^9} $ |
| 6 | $ 20a^4 n^{10} $ |
| 7 | $ 72n^7 r^7 $ |
| 8 | $ \frac{1}{k^{13}} $ |
| 9 | $ 90z^{14} $ |
| 10 | $ 9^9 $ |
| 11 | $ 54b^4 $ |
| 12 | $ 63s^6 $ |
| 13 | $ 16a^8 c^8 $ |
| 14 | $ s^8 $ |

All answers have only positive exponents, as required.

Let me know if you'd like this formatted as a printable worksheet!