The image you uploaded appears to be a worksheet summarizing the
Laws of Exponents. Below, I will explain each law and provide examples for clarity.
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1. Product of Powers
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Rule: $ a^m \cdot a^n = a^{m+n} $
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Explanation: When multiplying two powers with the same base, add the exponents.
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Example: $ 2^3 \cdot 2^4 = 2^{3+4} = 2^7 $
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2. Quotient of Powers
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Rule: $ \frac{a^m}{a^n} = a^{m-n} $
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Explanation: When dividing two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.
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Example: $ \frac{5^6}{5^2} = 5^{6-2} = 5^4 $
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3. Power of a Power
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Rule: $ (a^m)^n = a^{m \cdot n} $
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Explanation: When raising a power to another power, multiply the exponents.
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Example: $ (3^2)^4 = 3^{2 \cdot 4} = 3^8 $
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4. Power of a Product
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Rule: $ (ab)^m = a^m \cdot b^m $
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Explanation: When raising a product to a power, raise each factor to that power.
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Example: $ (xy)^3 = x^3 \cdot y^3 $
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5. Power of a Quotient
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Rule: $ \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m} $
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Explanation: When raising a quotient to a power, raise both the numerator and the denominator to that power.
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Example: $ \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9} $
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6. Negative Exponent
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Rule: $ a^{-n} = \frac{1}{a^n} $
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Explanation: A negative exponent indicates the reciprocal of the base raised to the positive exponent.
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Example: $ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} $
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7. Identity Exponent
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Rule: $ a^1 = a $
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Explanation: Any number raised to the power of 1 is the number itself.
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Example: $ 7^1 = 7 $
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8. Zero Exponent
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Rule: $ a^0 = 1 $ (where $ a \neq 0 $)
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Explanation: Any non-zero number raised to the power of 0 is 1.
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Example: $ 9^0 = 1 $
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Summary of Laws of Exponents:
| Name | Rule | Example |
|---------------------|-------------------------------|-----------------------|
| Product of Powers | $ a^m \cdot a^n = a^{m+n} $ | $ 2^3 \cdot 2^4 = 2^7 $ |
| Quotient of Powers | $ \frac{a^m}{a^n} = a^{m-n} $ | $ \frac{5^6}{5^2} = 5^4 $ |
| Power of a Power | $ (a^m)^n = a^{m \cdot n} $ | $ (3^2)^4 = 3^8 $ |
| Power of a Product | $ (ab)^m = a^m \cdot b^m $ | $ (xy)^3 = x^3 \cdot y^3 $ |
| Power of a Quotient | $ \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m} $ | $ \left(\frac{2}{3}\right)^2 = \frac{4}{9} $ |
| Negative Exponent | $ a^{-n} = \frac{1}{a^n} $ | $ 4^{-2} = \frac{1}{16} $ |
| Identity Exponent | $ a^1 = a $ | $ 7^1 = 7 $ |
| Zero Exponent | $ a^0 = 1 $ (where $ a \neq 0 $) | $ 9^0 = 1 $ |
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If you have a specific problem related to these laws, feel free to share it, and I can help solve it step by step!
Final Answer:
$$
\boxed{\text{See explanations above for each law of exponents.}}
$$
Parent Tip: Review the logic above to help your child master the concept of 7th grade math exponents worksheet.
