Exponent and root calculator

How to calculate exponents and roots

An exponent counts repeated multiplication: 14³ = 14 × 14 × 14 = 2,744. A root undoes a power, so the cube root of 1,728 is 12 because 12³ = 1,728. A fractional exponent combines the two, x^(m/n) = (ⁿ√x)^m, so 16^(2/3) = (∛16)² = 4∛4 ≈ 6.349604.

In 16^(2/3) the denominator 3 asks for a cube root and the numerator 2 for a square. That is different from 16² ÷ 3, which is about 85.33.

Squares, cubes and fractional powers

The exact form keeps a fraction or a simplified radical; the decimal column is rounded to six places, with whole numbers kept whole.

An exponent multiplies the base by itself; it never multiplies the base by the exponent. 2¹⁰ is 1,024, not 20, and 30³ is 27,000, not 90.

A decimal exponent is a fraction in disguise: 0.5 is 1/2, so 2^0.5 = √2. A negative exponent takes a reciprocal, which is why 0.25^(−1/2) = 1/√0.25 = 2.

Exponent rules

These rules hold for any positive base x:

• Product: xᵃ × xᵇ = xᵃ⁺ᵇ, so 2³ × 2⁴ = 2⁷ = 128.
• Quotient: xᵃ ÷ xᵇ = xᵃ⁻ᵇ, so 2⁶ ÷ 2² = 2⁴ = 16.
• Power of a power: (xᵃ)ᵇ = xᵃᵇ, so (2³)² = 2⁶ = 64.
• Power of a product: (xy)ᵃ = xᵃyᵃ, so (2 × 5)³ = 8 × 125 = 1,000.
• Zero exponent: x⁰ = 1, because xᵃ ÷ xᵃ = 1. By convention 0⁰ is also taken as 1.
• Negative exponent: x⁻ⁿ = 1/xⁿ, so 2⁻³ = 1/8 and 10⁻² = 1/100.
• Fractional exponent: x^(m/n) = (ⁿ√x)^m, so 8^(2/3) = (∛8)² = 4.

Zero to a positive power is 0, but zero to a negative power is undefined, because it means dividing by zero.

Cube roots and other roots

A perfect cube has a whole-number cube root: 216 = 6³, 343 = 7³ and 729 = 9³. Any other positive whole number has an irrational cube root, which simplifies by pulling out the largest perfect cube: 189 = 27 × 7, so ∛189 = 3∛7.

A square root of a square root is a fourth root: √(√2) = 2^(1/4) = ∜2 ≈ 1.189207, and √(√3) = ∜3. For square roots to hundreds of digits, use the square root calculator.

Negative bases and principal roots

Odd roots of negative numbers are real: ∛(−8) = −2 and the fifth root of −32 is −2. Even roots of negative numbers are not real numbers, so √(−16) needs the complex number calculator, where it is 4i.

A negative base with a fractional exponent stays real only when the fraction, in lowest terms, has an odd denominator. So (−8)^(2/3) = (∛(−8))² = 4 and (−8)^(−1/3) = −1/2.

The decimal 0.3333333333333333 is not 1/3. As a fraction its denominator is a power of ten, which is even, so −8 raised to it has no real value; type the exponent as 1/3.

The root sign names the principal root, the nonnegative one: √50 = 5√2 ≈ 7.071068. The equation x² = 50 has two solutions, ±5√2. OpenStax's radicals and rational-exponents chapter sets out these real-number rules.

Where powers show up

Squares and cubes convert area and volume units. Since 1 m = 100 cm, 1 m² = 100² = 10,000 cm² and 1 m³ = 100³ = 1,000,000 cm³.

Growth by a fixed rate is a power too. Money growing 5% a year for ten years is multiplied by 1.05¹⁰ ≈ 1.6289, and powers of ten carry the size of a number in scientific notation, as in 6.02 × 10²³.

Common exponent mistakes

• Dropping brackets on a negative base. −3² means −(3²) = −9, while (−3)² = 9.
• Distributing a power over a sum. (2 + 3)² = 25, not 2² + 3² = 13. The power-of-a-product rule works for products only.
• Adding exponents across different bases. 2³ × 3² = 8 × 9 = 72, not 6⁵ = 7,776. The product rule needs the same base.
• Typing a rounded fraction. 16^0.67 is not 16^(2/3). Enter the exponent as 2/3 to keep the exact cube root.

Entering powers and roots

• Base and exponent accept fractions such as 27/8 and 2/3, and expressions such as 16+9: base 16+9 with exponent 1/2 gives 5.
• A root index can be any whole number from 2 to 1,000.
• The exact form appears when the answer is a fraction or a simplified radical. An irrational input, such as pi, gives a decimal only.