How to Calculate Exponents and Powers
Learn how exponents and powers work, including the rules of exponents, negative and fractional exponents, and scientific notation. Step-by-step examples for every rule.
What Are Exponents?
An exponent (or power) tells how many times to multiply a base number by itself. In the expression 2⁵, the base is 2 and the exponent is 5, meaning 2 × 2 × 2 × 2 × 2 = 32. Exponents are fundamental in algebra, scientific notation, compound interest, and computer science.
The Product and Quotient Rules
When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 3² × 3⁴ = 3⁶ = 729. When dividing powers with the same base, subtract the exponents: aᵐ / aⁿ = aᵐ⁻ⁿ. So 5⁷ / 5³ = 5⁴ = 625.
The Power of a Power Rule
When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. For example, (2³)⁴ = 2¹² = 4,096. The power of a product rule states (ab)ⁿ = aⁿbⁿ, and the power of a quotient rule states (a/b)ⁿ = aⁿ/bⁿ.
Zero and Negative Exponents
Any nonzero number raised to the power of 0 equals 1: a⁰ = 1 (for a ≠ 0). A negative exponent indicates a reciprocal: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8. Negative exponents appear frequently in scientific notation for very small numbers like 6.02 × 10⁻²³.
Fractional Exponents and Radicals
A fractional exponent represents a root: a^(1/n) = ⁿ√a, and a^(m/n) = ⁿ√(aᵐ). For example, 8^(1/3) = ∛8 = 2, and 27^(2/3) = (∛27)² = 3² = 9. Fractional exponents unify the concepts of powers and roots into a single notation system.
Scientific Notation
Scientific notation expresses very large or very small numbers as a × 10ⁿ where 1 ≤ a < 10. The speed of light is approximately 3 × 10⁸ m/s, and an electron's mass is about 9.11 × 10⁻³¹ kg. To multiply numbers in scientific notation, multiply the coefficients and add the exponents of 10.
Common Mistakes to Avoid
A frequent error is applying exponent rules across addition: (a + b)² ≠ a² + b². The correct expansion is a² + 2ab + b². Another mistake is computing −3² as 9 instead of −9; since the exponent applies only to 3 (not the negative sign), −3² = −(3²) = −9, while (−3)² = 9.
Try These Calculators
Put what you learned into practice with these free calculators.
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