Index laws and rational powers

Explain the index laws using factors

For positive integer exponents, a^m aⁿ = a^(m + n) joins equal factors, (a^m)ⁿ = a^mn repeats a product, and a^m/aⁿ = a^(m-n) cancels common factors when a ≠ 0.

Extending consistently gives a⁰ = 1 and a⁻ⁿ = 1/aⁿ for a ≠ 0. A negative exponent makes a reciprocal, not a negative result. For positive a, a^(1/n) = nth root of a and a^(m/n) = (nth root of a)^m; using positive bases avoids real-domain ambiguity for fractional powers.

For 16^(3/4), the fourth root is 2, then cube it to obtain 8. For 25^(-1/2), the square root is 5, then take its reciprocal to obtain 1/5.

The base must match

2³ × 3³ = (2 × 3)³ = 216, but adding exponents as if the bases matched would be wrong. Also a² + a³ cannot be simplified to a⁵: the product law does not apply to addition.

Check your understanding 1 A cancellation needs its condition

Simplify x⁵/x² and state its domain. Then evaluate 8^(-2/3).

Your answer or drawing on paper

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The expression is x³ for x ≠ 0. The simplified polynomial is defined at zero, but the original quotient is not. Also 8^(-2/3) = 1/(cube root of 8)² = 1/4.