How Exponents and Powers Work
Exponentiation is the operation of raising a base number to a power (exponent). It represents repeated multiplication — 2³ means 2 × 2 × 2 = 8. Exponents extend beyond positive integers to include negative exponents (which produce fractions) and fractional exponents (which produce roots). Understanding exponents is essential for scientific notation, compound growth, and logarithms.
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Formula
Exponent / Power Calculator
Calculate base raised to any power, including fractional and negative exponents.
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Worked Example
Given:
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FAQs
What is a negative exponent?
A negative exponent means the reciprocal of the positive power: b⁻ⁿ = 1/bⁿ. For example, 2⁻³ = 1/2³ = 1/8 = 0.125. Negative exponents appear frequently in scientific notation for very small numbers, such as 10⁻⁹ = 0.000000001 (one nanometre).
What is a fractional exponent?
A fractional exponent represents a root: b^(1/n) = ⁿ√b. For example, 8^(1/3) = ∛8 = 2. Combined: b^(m/n) = (ⁿ√b)^m. So 8^(2/3) = (∛8)² = 2² = 4. Fractional exponents unify the concepts of powers and roots into a single notation.
Why does anything to the power of 0 equal 1?
By the pattern of dividing by the base: 2³=8, 2²=4, 2¹=2, 2⁰=1 (dividing 2 by 2). Also consistent with the exponent rule b^m ÷ b^m = b^(m-m) = b⁰, which must equal 1 because any number divided by itself is 1.