Exponent Calculator

Negative and fractional exponents

A negative exponent means a reciprocal: 2^-3 = 1 / 2^3 = 1/8 = 0.125. A fractional exponent means a root: 27^(1/3) is the cube root of 27, which is 3, and 16^(3/4) is the fourth root of 16 (which is 2) raised to the third power, giving 8.

These rules follow from the product rule a^m x a^n = a^(m+n). Since 2^3 x 2^-3 must equal 2^0 = 1, the value of 2^-3 has to be 1/8. In the same way, 9^(1/2) x 9^(1/2) = 9^1, so 9^(1/2) is the number that multiplies by itself to give 9, which is 3.

One caution: a negative base with a fractional exponent, such as (-8)^(1/2), has no real answer, because no real number squared gives -8. Odd roots work, though: (-8)^(1/3) = -2.

What an exponent means

An exponent tells you how many times to multiply a base by itself. In 2¹⁰, the base is 2 and the exponent is 10, so 2¹⁰ = 1,024. Any non-zero number to the power 0 equals 1, and any number to the power 1 is itself.

The exponent rules

Multiplying powers of the same base adds exponents: 2³ × 2⁴ = 2⁷ = 128. Dividing subtracts them: 5⁶ ÷ 5⁴ = 5² = 25. A power of a power multiplies them: (3²)³ = 3⁶ = 729.

A negative exponent means a reciprocal: 10⁻³ = 1 ÷ 10³ = 0.001. A fractional exponent means a root: 9^(1/2) = √9 = 3, and 8^(2/3) means the cube root of 8, squared, which is 2² = 4.

Note the order of operations: −3² = −9 because the exponent applies before the minus sign, while (−3)² = 9.

Exponential growth

Exponents describe anything that grows by a fixed percentage each period. Money growing at 8% a year for 10 years is multiplied by 1.08¹⁰ ≈ 2.16. A quantity that doubles 10 times grows by 2¹⁰, more than a thousand-fold, which is why exponential growth surprises people so often.