Exponent Calculator

How to use the exponent calculator

1. Enter the base. This is the number being multiplied. It can be negative or a decimal.
2. Enter the exponent. Type a whole number (10), a negative number (−3), a decimal (0.5) or a fraction (2/3). Fractions and decimals are treated as the same exponent, so 0.5 and 1/2 give the same result.
3. Read the result. The large figure is the power, written in full when it fits and in scientific notation when it doesn’t. The cards give the scientific-notation form, the number of digits for exact whole-number results, and the rule used for negative or fractional exponents.

The calculator opens on 2¹⁰ = 1,024. Try 2¹⁰⁰: the exact value has 31 digits, 1,267,650,600,228,229,401,496,703,205,376, which the calculator prints in full.

What an exponent means

An exponent tells you how many times to use the base as a factor. In aⁿ, a is the base and n is the exponent, or power. So 5³ = 5 × 5 × 5 = 125, and 2¹⁰ is ten 2s multiplied together.

Parentheses matter with negative bases. (−3)⁴ = 81, because four negative factors make a positive product, but −3⁴ means −(3⁴) = −81. The calculator treats the base you type as the whole number in parentheses, so a base of −3 with exponent 4 gives 81. A negative base raised to an odd whole-number power stays negative: (−3)³ = −27.

The exponent rules

OpenStax’s Algebra and Trigonometry 2e (section 1.2) lists the rules that make exponents easy to combine. For a nonzero base a:

The zero exponent rule follows from the quotient rule: t⁸ ÷ t⁸ is 1, and it is also t⁸⁻⁸ = t⁰. The negative rule works the same way: dividing by a larger power leaves the factors in the denominator.

Fractional exponents are roots

A fractional exponent combines a power and a root. Section 1.3 of the same textbook puts it this way: the numerator is the power and the denominator is the root.

In Example 11, 343^(2/3) is the cube root of 343, squared: ∛343 = 7, and 7² = 49. Taking the root first keeps the numbers small. An exponent of 1/2 is a square root, so 9^(1/2) = 3, and 0.5 works the same way.

Negative bases need care. If the denominator is odd, the root is real: (−32)^(1/5) = −2 and (−8)^(2/3) = 4. If the denominator is even, such as (−16)^(1/4), there is no real answer, and the calculator says so instead of returning an error code.

Very large and very small powers

Powers grow fast. 2¹⁰ is about a thousand, 2²⁰ about a million and 2³⁰ about a billion, which is why computer memory sizes come in powers of 2. Most calculators switch to scientific notation once a result passes about 10 to 15 digits and lose the trailing digits.

For a whole-number base and a positive whole-number exponent, this calculator multiplies exactly with arbitrary-length integers, so 3⁵⁰ or 7¹⁰⁰ comes out to the last digit. Other results are accurate to about 15 significant figures and shown in scientific notation when very large or very small. Negative exponents give small numbers: 10⁻² = 0.01 and 10⁻⁵ = 0.00001.

For converting between decimal and scientific notation, or multiplying numbers in that form, use the scientific notation calculator.

Where exponents appear

• Area and volume: a square 6 feet on a side covers 6² = 36 square feet; a cube 3 feet on a side holds 3³ = 27 cubic feet, which is one cubic yard.
• Doubling: a quantity that doubles 10 times grows by 2¹⁰ = 1,024 times, and one that doubles 20 times grows by more than a million.
• Science: powers of ten express sizes from 10⁻⁹ meters (a nanometer) to 10³ meters (a kilometer).
• Roots: exponents of 1/2 and 1/3 are square and cube roots, which the square root calculator also simplifies into radical form.

Common mistakes with exponents

• Multiplying the base by the exponent. 2³ is 2 × 2 × 2 = 8, not 2 × 3 = 6.
• Thinking a negative exponent makes a negative number. 2⁻³ is 1/8, a small positive number. The sign of the result depends on the base, not the exponent.
• Spreading a power over addition. (3 + 4)² = 7² = 49, but 3² + 4² = 9 + 16 = 25. Powers spread over multiplication, (ab)ⁿ = aⁿbⁿ, but not over addition.
• Adding exponents with different bases. 2³ × 3² cannot be combined into one power; it is 8 × 9 = 72. The product rule needs the same base.
• Multiplying exponents instead of adding. 2³ × 2⁴ = 2⁷, not 2¹². Exponents multiply only for a power of a power, (2³)⁴ = 2¹².
• Losing the sign of a negative base. On many phone calculators, typing −2 and then x² gives −4, because the minus is applied after squaring. Put the base in parentheses, (−2)² = 4, or use this calculator, which treats the base as one number.
• Treating a fractional exponent as division. 16^(1/2) is the square root of 16, which is 4, not 16 ÷ 2 = 8. Likewise 8^(1/3) is the cube root of 8, which is 2, not 8 ÷ 3.

Doubling and halving

Exponents describe anything that multiplies by the same factor over and over. A culture of bacteria that doubles every hour grows by a factor of 2ᵗ after t hours: 2⁵ = 32 times as many after five hours, 2¹⁰ = 1,024 after ten. Something that halves at each step shrinks by (1/2)ᵗ = 2⁻ᵗ: after three halvings, (1/2)³ = 1/8 of the original remains, which is how half-lives work in physics and medicine.

To go the other way and find the exponent, for example how many doublings it takes to pass 1,000, use a logarithm: log₂(1,000) ≈ 9.97, so ten doublings. The log calculator does this for any base.

Sources & assumptions

How this calculator gets its numbers. Full method: our methodology.

• a⁰ = 1 for a ≠ 0; 0⁰ is shown as 1, the convention used in algebra and most calculators.
• a⁻ⁿ = 1 ÷ aⁿ; a^(m/n) = (ⁿ√a)^m. A negative base with an even root in the exponent has no real value.
• Whole-number powers of whole numbers are computed exactly; other results to about 15 significant digits.

Sources

1. OpenStax (Rice University), Algebra and Trigonometry 2e, 1.2 Exponents and Scientific Notation. Product, quotient and power rules; the zero exponent rule a⁰ = 1 and the negative rule a⁻ⁿ = 1/aⁿ.
2. OpenStax (Rice University), Algebra and Trigonometry 2e, 1.3 Radicals and Rational Exponents. Rational exponents a^(m/n) = (ⁿ√a)^m: Example 11, 343^(2/3) = 49; Example 10, ⁵√−32 = −2.

Last reviewed November 2, 2026. Decimal exponents such as 0.5 are treated as the fraction they equal (1/2).