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Laws of Exponents. These laws are fundamental rules used to simplify expressions involving 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: \( a^m \times a^n = a^{m+n} \)
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Explanation: When multiplying two powers with the same base, add the exponents.
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Example: \( 4^2 \times 4^3 = 4^{2+3} = 4^5 \)
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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: \( \frac{7^5}{7^3} = 7^{5-3} = 7^2 \)
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3. Power of a Power
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Rule: \( (a^m)^n = a^{mn} \)
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Explanation: When raising a power to another power, multiply the exponents.
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Example: \( (2^3)^4 = 2^{3 \times 4} = 2^{12} \)
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4. Power of a Product
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Rule: \( (ab)^m = a^m \times b^m \)
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Explanation: When raising a product to a power, raise each factor to that power.
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Example: \( (3 \times 5)^4 = 3^4 \times 5^4 \)
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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: \( \left( \frac{1}{6} \right)^2 = \frac{1^2}{6^2} = \frac{1}{6^2} \)
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6. Zero Exponent
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Rule: \( a^0 = 1 \) (where \( a \neq 0 \))
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Explanation: Any nonzero number raised to the power of 0 is equal to 1.
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Example: \( 8^0 = 1 \)
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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: \( 5^1 = 5 \)
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8. 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: \( 9^{-1} = \frac{1}{9} \)
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Summary of the Laws:
| Law | Rule | Example |
|--------------------|-------------------------------|-----------------------|
| Product of Powers | \( a^m \times a^n = a^{m+n} \) | \( 4^2 \times 4^3 = 4^5 \) |
| Quotient of Powers | \( \frac{a^m}{a^n} = a^{m-n} \) | \( \frac{7^5}{7^3} = 7^2 \) |
| Power of a Power | \( (a^m)^n = a^{mn} \) | \( (2^3)^4 = 2^{12} \) |
| Power of a Product | \( (ab)^m = a^m \times b^m \) | \( (3 \times 5)^4 = 3^4 \times 5^4 \) |
| Power of a Quotient| \( \left( \frac{a}{b} \right)^m = \frac{a^m}{b^m} \) | \( \left( \frac{1}{6} \right)^2 = \frac{1}{6^2} \) |
| Zero Exponent | \( a^0 = 1 \) (where \( a \neq 0 \)) | \( 8^0 = 1 \) |
| Identity Exponent | \( a^1 = a \) | \( 5^1 = 5 \) |
| Negative Exponent | \( a^{-n} = \frac{1}{a^n} \) | \( 9^{-1} = \frac{1}{9} \) |
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These laws are essential for simplifying and solving problems involving exponents. If you have any specific questions or need further clarification, feel free to ask!
Final Answer:
\[
\boxed{\text{See explanations above}}
\]
Parent Tip: Review the logic above to help your child master the concept of basic exponent rules worksheet.
