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DayUtils

Exponent Calculator – Powers and Roots of Any Number

Raise a number to any power — positive, negative or fractional — or find its square root, cube root or any nth root. The answer is shown in full, in scientific notation and with every digit written out, so even huge results are readable.

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How to use the exponent calculator

  1. Choose Power xⁿ to raise a number to a power, or Root ⁿ√x to take a root.
  2. Enter the base (or the number for a root).
  3. Enter the exponent (or which root: 2 for square, 3 for cube).

You’ll see the result, its scientific notation and every digit written out. For small whole-number powers, the calculator also shows the repeated multiplication.

Exponent rules

RuleExample
xᵃ × xᵇ = xᵃ⁺ᵇ2³ × 2² = 2⁵ = 32
xᵃ ÷ xᵇ = xᵃ⁻ᵇ5⁴ ÷ 5² = 5² = 25
(xᵃ)ᵇ = xᵃᵇ(3²)³ = 3⁶ = 729
x⁰ = 17⁰ = 1
x⁻ᵃ = 1 ÷ xᵃ10⁻² = 0.01
x^(1/n) = ⁿ√x27^(1/3) = 3
(xy)ᵃ = xᵃyᵃ(2 × 5)³ = 1,000

Worked examples

  • 2^10 = 2 × 2 × … × 2 (ten times) = 1,024
  • 10^−3 = 1 ÷ 1,000 = 0.001
  • 9^0.5 = √9 = 3
  • 8^(2/3) = (∛8)² = 2² = 4
  • Cube root of −27 = −3 (because −3 × −3 × −3 = −27)
  • 1.05^10 ≈ 1.6289 — what £1 grows to in 10 years at 5% compound interest

Where powers are used

  • Compound growth: savings, inflation and population growth all use (1 + rate)^years. Try the compound interest calculator.
  • Area and volume: the area of a square is side², the volume of a cube is side³.
  • Science: very large and small quantities are written as powers of 10 — see the scientific notation converter.
  • Computing: memory and storage sizes are powers of 2.
  • Finance and physics formulas often contain squares, roots and exponents.

Very large and very small results

Powers grow extremely fast. 2^100 is more than a million trillion trillion. When a result is larger than about 10²¹ or smaller than 0.000001, it is shown in scientific notation, and the full-digit version is written out underneath where possible. Results bigger than about 1.8 × 10³⁰⁸ are too large for the calculator to represent.

Common mistakes with exponents

  • Negative bases need brackets. (−2)² = 4, but −2² means −(2²) = −4.
  • Powers don’t spread over addition. (a + b)² is a² + 2ab + b², not a² + b².
  • A negative exponent doesn’t make a number negative. 2⁻³ is 0.125, a small positive number.
  • Multiplying powers adds the exponents only when the base is the same: 2³ × 3² can’t be combined that way.

Exponents and logarithms

A logarithm undoes an exponent. If 2⁵ = 32, then log₂(32) = 5 — it answers “what power do I need?”. Use the scientific calculator for log(x, b).

Squares and square roots to remember

1² = 1, 2² = 4, 3² = 9 … 10² = 100, 12² = 144, 15² = 225, 20² = 400, 25² = 625. Knowing these makes it easier to estimate roots: √50 is a little over 7 because 7² = 49.

For expressions with several operations, use the scientific calculator. For factors and multiples, try the LCM and GCF calculator.

Frequently asked questions

What is an exponent?

An exponent tells you how many times to multiply a number by itself. In 2^5, 2 is the base and 5 the exponent: 2 × 2 × 2 × 2 × 2 = 32.

What does a negative exponent mean?

It means one divided by the positive power. 2^−3 = 1 ÷ 2³ = 1/8 = 0.125.

What does a fractional exponent mean?

A fractional exponent is a root. x^(1/2) is the square root of x, x^(1/3) the cube root, and x^(2/3) the cube root of x, squared. 8^(2/3) = 4.

What is anything to the power of 0?

Any non-zero number raised to the power 0 equals 1. By convention, 0^0 is also treated as 1 in most calculators.

Can I find the root of a negative number?

Odd roots of negative numbers are real: the cube root of −27 is −3. Even roots, such as the square root of −4, aren’t real numbers, so the calculator shows a message instead.

What is 2 to the power 10?

1,024. Powers of 2 are important in computing: 2^10 = 1,024 is why a kibibyte is 1,024 bytes.

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