Understanding exponent rules is key in Algebra 1, making calculations easier and more efficient. These rules help simplify expressions involving powers, whether you're multiplying, dividing, or dealing with negative and fractional exponents. Let's break them down!
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Product Rule: am × an = am+n
- When multiplying two expressions with the same base, add their exponents.
- This rule simplifies calculations involving powers of the same base.
- Example: 2^3 × 2^2 = 2^(3+2) = 2^5 = 32.
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Quotient Rule: am ÷ an = am-n
- When dividing two expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator.
- This rule helps simplify fractions involving powers.
- Example: 5^4 ÷ 5^2 = 5^(4-2) = 5^2 = 25.
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Power of a Power Rule: (am)n = amn
- When raising a power to another power, multiply the exponents.
- This rule is useful for simplifying expressions with multiple layers of exponents.
- Example: (3^2)^4 = 3^(2×4) = 3^8 = 6561.
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Power of a Product Rule: (ab)n = anbn
- When raising a product to a power, distribute the exponent to each factor in the product.
- This rule allows for easier manipulation of products raised to powers.
- Example: (2 × 3)^3 = 2^3 × 3^3 = 8 × 27 = 216.
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Zero Exponent Rule: a0 = 1
- Any non-zero base raised to the power of zero equals one.
- This rule is fundamental in understanding the behavior of exponents.
- Example: 7^0 = 1.
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Negative Exponent Rule: a-n = 1/an
- A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent.
- This rule helps in simplifying expressions with negative exponents.
- Example: 2^-3 = 1/(2^3) = 1/8.
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Fractional Exponent Rule: a1/n = nth root of a
- A fractional exponent represents a root; the numerator is the power and the denominator is the root.
- This rule connects exponents with radical expressions.
- Example: 16^(1/2) = √16 = 4.