Laws of Exponents exercise for 7 - Free Printable

Let's solve each problem step by step using the Laws of Exponents provided:

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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^m}{b^n}\right)^p = \frac{a^{mp}}{b^{np}} $

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Now, let’s simplify each expression.

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1. $ a^3 \cdot a^5 $

Use Law 1: $ a^m \cdot a^n = a^{m+n} $

$$
a^{3+5} = a^8
$$

✔ Answer: $ a^8 $

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2. $ a^3b \cdot ab^6 $

Group like terms:

- $ a^3 \cdot a = a^{3+1} = a^4 $
- $ b \cdot b^6 = b^{1+6} = b^7 $

So: $ a^4b^7 $

✔ Answer: $ a^4b^7 $

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3. $ (2b^5)^3 $

Apply Law 4: $ (ab)^n = a^n b^n $

$$
= 2^3 \cdot (b^5)^3 = 8 \cdot b^{15}
$$

✔ Answer: $ 8b^{15} $

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4. $ (a^4b^2)^3 $

Apply Law 4: $ (ab)^n = a^n b^n $

$$
= (a^4)^3 \cdot (b^2)^3 = a^{12} \cdot b^6
$$

✔ Answer: $ a^{12}b^6 $

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5. $ (a^2)^3 (a^2)^2 $

Use Law 3: $ (a^m)^n = a^{mn} $

- $ (a^2)^3 = a^{6} $
- $ (a^2)^2 = a^{4} $

Now multiply: $ a^6 \cdot a^4 = a^{10} $

✔ Answer: $ a^{10} $

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6. $ (3b^2)(2b) $

Multiply coefficients and variables:

- Coefficients: $ 3 \cdot 2 = 6 $
- Variables: $ b^2 \cdot b = b^{3} $

✔ Answer: $ 6b^3 $

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7. $ \frac{a^7}{a^3} $

Use Law 2: $ \frac{a^m}{a^n} = a^{m-n} $

$$
a^{7-3} = a^4
$$

✔ Answer: $ a^4 $

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8. $ \left( \frac{a^3b^5}{ab} \right)^2 $

First simplify inside the parentheses:

- $ \frac{a^3}{a} = a^{3-1} = a^2 $
- $ \frac{b^5}{b} = b^{5-1} = b^4 $

So: $ \frac{a^3b^5}{ab} = a^2b^4 $

Now square it: $ (a^2b^4)^2 = a^{4}b^{8} $

✔ Answer: $ a^4b^8 $

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9. $ \frac{(4a^4)^2}{(2a)^3} $

Simplify numerator and denominator separately.

Numerator: $ (4a^4)^2 = 4^2 \cdot (a^4)^2 = 16a^8 $

Denominator: $ (2a)^3 = 2^3 \cdot a^3 = 8a^3 $

Now divide:

$$
\frac{16a^8}{8a^3} = \frac{16}{8} \cdot a^{8-3} = 2a^5
$$

✔ Answer: $ 2a^5 $

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10. $ (5a^2bc^4)^0 $

Any non-zero expression raised to the power of 0 is 1.

Even though it has variables, as long as the base is not zero, $ x^0 = 1 $

✔ Answer: $ 1 $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1. | $ a^8 $ |
| 2. | $ a^4b^7 $ |
| 3. | $ 8b^{15} $ |
| 4. | $ a^{12}b^6 $ |
| 5. | $ a^{10} $ |
| 6. | $ 6b^3 $ |
| 7. | $ a^4 $ |
| 8. | $ a^4b^8 $ |
| 9. | $ 2a^5 $ |
| 10. | $ 1 $ |

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📝 Tips to Remember:

- Always group like bases.
- Apply exponent rules carefully: add exponents when multiplying same bases, subtract when dividing.
- When raising a product to a power, apply the exponent to each factor.
- Any nonzero number to the power of 0 is 1.

You did great! 😊
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