Exponent and Root Calculator with Steps

What are exponents and roots?

An exponent counts repeated multiplication, so 3⁴ means 3 × 3 × 3 × 3 = 81. A root does the reverse: the square root of 81 is 9, because 9² = 81, and the fourth root of 81 is 3. Roots can also be written as fractional exponents, so the nth root of a equals a to the power 1/n.

Exponent rules

Rule	Example
Product: aᵐ × aⁿ = aᵐ⁺ⁿ	2³ × 2⁴ = 2⁷ = 128
Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ	5⁶ ÷ 5⁴ = 5² = 25
Power of a power: (aᵐ)ⁿ = aᵐⁿ	(3²)³ = 3⁶ = 729
Zero: a⁰ = 1 (a ≠ 0)	7⁰ = 1
Negative: a⁻ⁿ = 1 ÷ aⁿ	5⁻² = 1 ÷ 25 = 0.04
Fractional: a^(1/n) is the nth root	27^(1/3) = 3

Worked examples

A positive power

2⁵ = 2 × 2 × 2 × 2 × 2 = 32, and 10³ = 1,000.

A negative power

5⁻² = 1 ÷ 5² = 1 ÷ 25 = 0.04.

A cube root

The cube root of 27 is 3, because 3 × 3 × 3 = 27. The cube root of −8 is −2, since (−2)³ = −8.

Simplifying a radical

For √72, the largest perfect square factor is 36, so √72 = √(36 × 2) = 6√2, which is about 8.485.

Square roots, cube roots and nth roots

The square root of a number is the value that multiplies by itself to give it, and the cube root multiplies three times. Even roots of negative numbers are not real, while odd roots are, because an odd number of negative factors stays negative. By convention, the root symbol gives the non-negative (principal) root, so √25 is 5, not −5.

Common exponent and root mistakes

The expression −3² means −(3²) = −9, while (−3)² = 9, so brackets matter. Adding exponents only works when the bases match: 2³ × 3² is not 6⁵. A root does not split over addition, so √(9 + 16) = 5, not 3 + 4 = 7. And 0⁰ has no single agreed value. For exponents inside scientific problems, check the percentage calculator for growth rates, or the Pythagorean theorem calculator, which uses square roots.

Frequently asked questions

What does an exponent mean?

An exponent tells you how many times to multiply a number by itself. In 2⁵, the base 2 is multiplied five times: 2 × 2 × 2 × 2 × 2 = 32. Exponents give a short way to write repeated multiplication and very large or small numbers.

Why is any number to the power 0 equal to 1?

Dividing aⁿ by aⁿ always gives 1, and the quotient rule says it also equals a to the power n − n, which is a⁰. So a⁰ must be 1 for any non-zero a. The case 0⁰ is treated as undefined or context-dependent.

What does a negative exponent mean?

A negative exponent means the reciprocal of the positive power: a⁻ⁿ = 1 ÷ aⁿ. So 5⁻² = 1 ÷ 25 = 0.04. The minus sign does not make the answer negative; it moves the base to the other side of a fraction.

Can you take the square root of a negative number?

Not within the real numbers. No real number multiplied by itself gives a negative result, so √(−4) is not real. It is a complex number, written 2i. Odd roots are different: the cube root of −8 is the real number −2.

How do you simplify a square root?

Find the largest perfect square that divides the number, then take its root outside the radical. For 72, the largest perfect square factor is 36, so √72 = √(36 × 2) = 6√2. If no square factor exists, the root is already simplest.

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Related topics on this page: square root, cube root, nth root, powers of a number, negative exponents, exponent rules, simplify radicals, perfect squares