Exponent Calculator

This exponent calculator solves bⁿ = x for any one of its three parts. In Power mode it raises a base b to an exponent n. The other two modes find the base (an nth root) or the exponent (a logarithm). Bases and exponents can be whole numbers, decimals, fractions or negative numbers. You get the exact answer when one exists, such as 2¹⁰⁰ in full, 8^(2/3) = 4 or 12^(3/2) = 24√3, a decimal value with up to 100 significant digits, and the steps.

What an exponent means

For a whole number n, bⁿ is the product of n copies of b: 3⁴ = 3 × 3 × 3 × 3 = 81. The number b is the base and n is the exponent, or power. All other cases follow from one rule, bᵐ · bⁿ = bᵐ⁺ⁿ. If this rule must hold for n = 0, then b⁰ = 1. For a negative exponent it gives b⁻ⁿ = 1/bⁿ. For fractions, (b^(1/q))^q = b, so b^(1/q) is the qth root of b, and b^(p/q) is the qth root of bᵖ. For an irrational exponent such as π, bⁿ is defined as e^(n · ln b) when b > 0.

How the calculator works

Whole-number exponents. Results up to 20,000 digits are computed exactly with integer arithmetic, so 2¹⁰⁰ = 1,267,650,600,228,229,401,496,703,205,376 is shown in full. The calculator uses repeated squaring: since 100 = 64 + 32 + 4, 2¹⁰⁰ = 2⁶⁴ × 2³² × 2⁴, which needs only 8 multiplications instead of 99. Larger results, such as 2^1,000,000, are shown in scientific notation together with the number of digits and the exact last 20 digits, which are found by working modulo 10²⁰.

Fractional and decimal exponents. A decimal exponent is first written as a fraction in lowest terms, so 1.5 = 3/2. Then b^(p/q) is the qth root of bᵖ, and the calculator simplifies the root by taking out perfect qth powers: 12^(3/2) = √1728 = 24√3, because 1728 = 24² × 3. The decimal value comes from e^(n · ln b). The logarithm and the exponential are computed with extra guard digits beyond the precision you choose, and the last digit shown is rounded.

Negative bases. (−2)⁴ = 16 and (−2)³ = −8: an even exponent removes the minus sign and an odd exponent keeps it. For fractional exponents, the calculator follows the real-root convention of algebra textbooks. If the reduced fraction has an odd denominator, the root is real: (−8)^(1/3) = −2 and (−8)^(2/3) = 4. If the denominator is even, there is no real answer, so the calculator gives the principal complex value: (−4)^(1/2) = 2i.

Finding the base or the exponent. To solve bⁿ = x for b, raise both sides to the power 1/n: b = x^(1/n). An even exponent gives two real answers, such as b = ±3 for b⁴ = 81. To solve for n, take logarithms: n = ln x / ln b. When n is a fraction, the calculator finds it exactly and confirms it with integer arithmetic. For example, log₈ 4 = 2/3 because 8^(2/3) = 4. Otherwise it gives n to the chosen number of significant digits, for example log₂ 10 ≈ 3.32192809488736.

Worked examples

• 2¹⁰ = 1,024: ten factors of 2. This is why a kibibyte is 1,024 bytes.
• 5⁻³ = 1/125 = 0.008: a negative exponent gives the reciprocal.
• 27^(2/3) = 9: the cube root of 27 is 3, and 3² = 9.
• 2^0.5 = √2 ≈ 1.41421356237310: an exponent of 1/2 is a square root.
• 1.05¹⁰ ≈ 1.62889462677744: money that grows 5% a year for 10 years grows by about 62.9% in total.
• b³ = −125 gives b = −5, and 2ⁿ = 1,000,000 gives n ≈ 19.9315685693242.

Where exponents are used

• Compound interest: an amount P at a yearly rate r grows to P(1 + r)ᵗ after t years.
• Scientific notation: very large and very small numbers are written with powers of 10, such as 6.022 × 10²³ for Avogadro's number.
• Computing: n bits can store 2ⁿ different values, so 32 bits give 2³² = 4,294,967,296.
• Growth and decay: a population that doubles every period grows like 2ᵗ, and a radioactive sample keeps (1/2)^(t/h) of its mass after time t, where h is the half-life.

For roots on their own, the Square Root Calculator and the Root Calculator in this category give more detail on simplifying radicals.