Radicals and Integer Exponents

An exponent tells how many times a base is used as a factor. In 2⁴, 2 is the base and 4 is the exponent: 2⁴ = 2 × 2 × 2 × 2 = 16. A negative base must be grouped: (−3)² = 9, while −3² means −(3²) = −9.

For the same nonzero base: aᵐ · aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ. A power of a product distributes to each factor: (ab)ⁿ = aⁿbⁿ. These rules do not apply to addition: (a + b)² is not a² + b².

For any nonzero a, a⁰ = 1. A negative exponent means reciprocal: a⁻ⁿ = 1/aⁿ. For example, 5⁻² = 1/25.

A radical symbol indicates a root. √49 = 7 because 7² = 49. ∛64 = 4 because 4³ = 64. Square roots and squaring are inverse operations, but √x² = |x| because the principal square root is nonnegative.

Worked example: Simplify (x³)(x⁵)/x². Add exponents when multiplying and subtract when dividing: x³⁺⁵⁻² = x⁶, for x ≠ 0.