Exponent Calculator — Any Base to Any Power

b^n = b × b × b ... (n times)

b^0 = 1 (any non-zero base)
b^−n = 1/b^n (negative exponent = reciprocal)
b^(1/n) = nth root of b
b^(m/n) = (nth root of b)^m

Key rules:
b^m × b^n = b^(m+n)
(b^m)^n = b^(m×n)
b^m / b^n = b^(m−n)

Exponent rules follow from the definition of repeated multiplication. Understanding why the rules work — not just that they work — makes simplifying expressions much faster.

b^n (base A1, exponent B1)

=A1^B1

Scientific notation result

=TEXT(A1^B1,"0.00E+00")

Exponents and Powers ExplainedSearch YouTube tutorials →

Understanding Exponents

An exponent indicates how many times a base number is multiplied by itself. In 2^8, the base is 2 and the exponent is 8: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256. Exponents provide compact notation for repeated multiplication, just as multiplication provides compact notation for repeated addition.

The second power is called "squared" because it gives the area of a square with that side length. The third power is "cubed" because it gives a cube's volume. Exponents appear throughout finance (compound interest: P × (1+r)^n), physics (radioactive decay, energy equations), computer science (binary numbers: 2^10 = 1,024), and statistics (normal distribution formula).

Special cases worth knowing: any non-zero number to the 0 power equals 1. Any number to the 1st power equals itself. Negative exponents produce reciprocals: 2^(-3) = 1/8. Fractional exponents produce roots: 8^(1/3) = cube root of 8 = 2.

The Core Exponent Rules

All exponent rules derive logically from the definition of exponents as repeated multiplication. Product rule: b^m × b^n = b^(m+n). This is because multiplying two groups of repeated multiplication combines them: 2^3 × 2^4 = (2×2×2) × (2×2×2×2) = 2^7.

Power rule: (b^m)^n = b^(m×n). Raising a power to a power repeats the multiplication again: (2^3)^4 = 2^3 × 2^3 × 2^3 × 2^3 = 2^12. Quotient rule: b^m / b^n = b^(m-n). Dividing cancels factors: 2^7 / 2^3 = (2×2×2×2×2×2×2) / (2×2×2) = 2^4.

Frequently Asked Questions

A negative exponent means take the reciprocal: b^(-n) = 1/b^n. So 2^(-3) = 1/2^3 = 1/8 = 0.125. Negative exponents appear in scientific notation for very small numbers (10^(-9) = one billionth) and in physics formulas involving inverse-square laws.

b^(1/n) means the nth root of b. b^(1/2) = square root. b^(1/3) = cube root. b^(m/n) = the nth root of b, raised to the m power. So 8^(2/3) = (cube root of 8)^2 = 2^2 = 4. Fractional exponents unify the concepts of roots and powers under one algebraic framework.

Any non-zero number to the power of 0 equals 1. The intuitive proof: b^n / b^n = b^(n-n) = b^0, and any number divided by itself equals 1. Therefore b^0 = 1. The expression 0^0 is mathematically indeterminate (its value depends on context) and is generally either left undefined or defined as 1 for computational convenience.

Compound interest formula: A = P(1 + r)^n, where A = final amount, P = principal, r = rate per period, n = number of periods. The exponent n creates exponential growth — each period, the previous balance multiplies by (1+r). This is why compound interest is so powerful over long time horizons and why the Rule of 72 (72 / rate = years to double) works.