Exponents and Radicals Worksheets | Exponents & Radicals ... - Free Printable

Let's solve each problem step by step using the laws of exponents. Remember these key rules:

- Product of powers: $ a^m \cdot a^n = a^{m+n} $
- Quotient of powers: $ \frac{a^m}{a^n} = a^{m-n} $
- Power of a power: $ (a^m)^n = a^{m \cdot n} $
- Negative exponent: $ a^{-n} = \frac{1}{a^n} $, so we want to eliminate negative exponents and write answers with positive exponents only.

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1) $ \frac{h}{h^2} $

$$
= h^{1 - 2} = h^{-1} = \frac{1}{h}
$$

✔ Answer: $ \frac{1}{h} $

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2) $ \frac{d^7}{d^5} $

$$
= d^{7 - 5} = d^2
$$

✔ Answer: $ d^2 $

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3) $ c^5 \cdot c^6 \cdot c^4 $

$$
= c^{5+6+4} = c^{15}
$$

✔ Answer: $ c^{15} $

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4) $ \frac{8bg^2}{8b^{-2}g^4} $

Break it down:
- Coefficients: $ \frac{8}{8} = 1 $
- $ b $: $ \frac{b}{b^{-2}} = b^{1 - (-2)} = b^{3} $
- $ g $: $ \frac{g^2}{g^4} = g^{2-4} = g^{-2} = \frac{1}{g^2} $

So:
$$
= b^3 \cdot \frac{1}{g^2} = \frac{b^3}{g^2}
$$

✔ Answer: $ \frac{b^3}{g^2} $

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5) $ \frac{2b^6n^8}{5bn^3} $

Break it down:
- Coefficients: $ \frac{2}{5} $
- $ b $: $ \frac{b^6}{b} = b^{6-1} = b^5 $
- $ n $: $ \frac{n^8}{n^3} = n^{8-3} = n^5 $

$$
= \frac{2}{5} b^5 n^5
$$

✔ Answer: $ \frac{2}{5} b^5 n^5 $

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6) $ \frac{9zd^4}{4z^2d^5} $

Break it down:
- Coefficients: $ \frac{9}{4} $
- $ z $: $ \frac{z}{z^2} = z^{1-2} = z^{-1} = \frac{1}{z} $
- $ d $: $ \frac{d^4}{d^5} = d^{-1} = \frac{1}{d} $

So:
$$
= \frac{9}{4} \cdot \frac{1}{z} \cdot \frac{1}{d} = \frac{9}{4zd}
$$

✔ Answer: $ \frac{9}{4zd} $

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

Multiply coefficients: $ 2 \cdot 4 = 8 $

Multiply variables: $ d \cdot d^2 = d^{1+2} = d^3 $

$$
= 8d^3
$$

✔ Answer: $ 8d^3 $

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8) $ 5k^3 \cdot 9k^2 $

Coefficients: $ 5 \cdot 9 = 45 $

Variables: $ k^3 \cdot k^2 = k^{3+2} = k^5 $

$$
= 45k^5
$$

✔ Answer: $ 45k^5 $

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9) $ \frac{k^2}{k^{-4}} $

$$
= k^{2 - (-4)} = k^{2+4} = k^6
$$

✔ Answer: $ k^6 $

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10) $ z^3w^{-6} \cdot 8z^6w^4 \cdot 4zw^5 $

Group like terms:

- Coefficients: $ 1 \cdot 8 \cdot 4 = 32 $
- $ z $: $ z^3 \cdot z^6 \cdot z = z^{3+6+1} = z^{10} $
- $ w $: $ w^{-6} \cdot w^4 \cdot w^5 = w^{-6+4+5} = w^3 $

$$
= 32z^{10}w^3
$$

✔ Answer: $ 32z^{10}w^3 $

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11) $ wb \cdot 6w^6b^4 $

Coefficients: $ 1 \cdot 6 = 6 $

$ w \cdot w^6 = w^{1+6} = w^7 $

$ b \cdot b^4 = b^{1+4} = b^5 $

$$
= 6w^7b^5
$$

✔ Answer: $ 6w^7b^5 $

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12) $ 4c^3 \cdot 5c^4 $

Coefficients: $ 4 \cdot 5 = 20 $

Variables: $ c^3 \cdot c^4 = c^{3+4} = c^7 $

$$
= 20c^7
$$

✔ Answer: $ 20c^7 $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $ \frac{1}{h} $ |
| 2) | $ d^2 $ |
| 3) | $ c^{15} $ |
| 4) | $ \frac{b^3}{g^2} $ |
| 5) | $ \frac{2}{5} b^5 n^5 $ |
| 6) | $ \frac{9}{4zd} $ |
| 7) | $ 8d^3 $ |
| 8) | $ 45k^5 $ |
| 9) | $ k^6 $ |
| 10) | $ 32z^{10}w^3 $ |
| 11) | $ 6w^7b^5 $ |
| 12) | $ 20c^7 $ |

All answers have only positive exponents, as required.