An exponent tells you how many times to multiply a base number by itself. For example, 2³ means 2 × 2 × 2 = 8. The base is 2 and the exponent (or power) is 3.

How do I calculate a square root using this calculator?

Enter the number as the base, then enter 0.5 as the exponent (or type 1/2). The square root is the same as raising to the power of 0.5. For example, 9^0.5 = 3.

What does a negative exponent mean?

A negative exponent means take the reciprocal. For example, 2⁻³ = 1 ÷ 2³ = 1/8 = 0.125. In general, a⁻ⁿ = 1/aⁿ.

What is a fractional exponent?

A fractional exponent like a^(m/n) means the nth root of aᵐ. For example, 8^(1/3) = ∛8 = 2, and 4^(3/2) = (√4)³ = 2³ = 8.

What is zero raised to the power of zero?

0⁰ is mathematically indeterminate — it is undefined in the general sense, though some contexts define it as 1 by convention (particularly in combinatorics and polynomial evaluation).

What is the product rule for exponents?

When multiplying two powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 2³ × 2⁴ = 2⁷ = 128.

How does the Growth & Decay table work?

Enter a base (multiplier), a starting value, and the number of periods. The table shows how the value changes each period by multiplying by the base. A base > 1 shows growth; a base between 0 and 1 shows decay. Use presets for common scenarios like compound interest or radioactive half-life.

Can this calculator handle very large numbers?

Yes. For very large results (greater than 1,000,000) or very small results (less than 0.001), the calculator automatically displays the result in scientific notation alongside the standard form.

Exponent Calculator — Powers, Roots & Fractional Exponents

How to Use the Exponent Calculator

Choose a mode

Select bⁿ to raise a base to a power, ⁿ√b to take the nth root, or b^(1/n) to compute a unit-fraction exponent.

Enter the base and exponent

Type any real number for the base. The exponent accepts integers (5), decimals (0.5), and fractions (1/3).

Read the result

The result updates instantly. For very large or small numbers, scientific notation appears automatically. The step-by-step panel explains how the answer was reached.

Explore rules & growth

Switch to the Exponent Rules tab to see all six rules with your own numbers, or use the Growth & Decay tab to visualize exponential sequences.

Key Terms

Base

The number being raised to a power. In 5³, the base is 5.

Exponent / Power

The number indicating how many times to multiply the base. In 5³, the exponent is 3.

Radical / Root

The inverse of a power. The nth root of a is written ⁿ√a or a^(1/n).

Scientific Notation

A way to express very large or small numbers as a × 10ⁿ (e.g. 1.024 × 10³).

Exponential Growth

When a quantity multiplies by a fixed factor each period (factor > 1).

Exponential Decay

When a quantity multiplies by a fixed factor each period where 0 < factor < 1.

Worked Examples

2¹⁰ — Large integer power

2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024

This is why computer memory is measured in powers of 2 (1 KB = 2¹⁰ bytes).

8^(1/3) — Cube root

8^(1/3) = ∛8 = 2

Because 2 × 2 × 2 = 8. Fractional exponent 1/n gives the nth root.

(−3)⁴ — Negative base, even exponent

(−3)⁴ = (−3) × (−3) × (−3) × (−3) = 81

An even exponent on a negative base always gives a positive result.

5⁻² — Negative exponent

5⁻² = 1 / 5² = 1 / 25 = 0.04

Negative exponents move the power to the denominator.

4^(3/2) — Fractional exponent m/n

4^(3/2) = (√4)³ = 2³ = 8

Compute the root first, then raise to the power, for easier arithmetic.

10⁻³ — Scientific context

10⁻³ = 0.001

Powers of 10 define the metric prefix system: milli- = 10⁻³, kilo- = 10³.

Exponents in the Real World

Finance & Compound Interest

Compound interest is the most practical application of exponents. When money grows at rate r for n periods, the balance is A = P(1 + r)ⁿ. At 7% annual return over 30 years, $10,000 becomes $10,000 × 1.07³⁰ ≈ $76,123 — a nearly 8× increase. This dramatic result comes entirely from the exponent; small changes in r or n produce large changes in the outcome.

Science & Nature

Radioactive decay follows A = A₀ × (1/2)^(t/T½), where T½ is the half-life. After 3 half-lives, only (1/2)³ = 1/8 of the original material remains. Bacteria double roughly every 20 minutes under ideal conditions, so after t minutes a colony is n₀ × 2^(t/20). Exponents also appear in the Richter scale (each unit = 10× energy), decibel scale, and pH (each unit = 10× hydrogen concentration).

Computing & Information

Binary numbers are sums of powers of 2. The byte size of a file, the address space of a CPU, and the capacity of RAM are all expressed in powers of 2 (kilobyte = 2¹⁰, megabyte = 2²⁰, gigabyte = 2³⁰). Moore's Law famously predicted transistor counts doubling every ~2 years — an exponential trend that held for 50 years.

Why Fractional Exponents Matter

Fractional exponents unify powers and roots into a single notation. a^(1/2) = √a, a^(1/3) = ∛a, and a^(2/3) = (∛a)². This lets you apply all the standard power rules (product, quotient, chain) to roots as well, which is essential in algebra, calculus, and physics.

Frequently Asked Questions

What is the difference between a power and an exponent?

They refer to the same thing. "Exponent" describes the number in the superscript position (the 3 in 2³). "Power" describes the entire expression (2³ is called "the third power of 2" or "2 to the power of 3"). In casual use the terms are interchangeable.

What is 0⁰?

0⁰ is mathematically indeterminate — analysis gives conflicting answers (0ⁿ → 0, but n⁰ → 1). Discrete math conventionally uses 0⁰ = 1 for combinatorics and polynomials, but calculus treats it as undefined. This calculator displays a note rather than a number.

Can I raise a negative number to a fractional exponent?

Over the real numbers, (−8)^(1/3) = −2 is valid (cube root of −8), but (−4)^(1/2) is undefined in the reals (square root of a negative number). This calculator notes when the result would be complex.

How is the step-by-step solution determined?

For small integer exponents (≤ 12), the calculator shows repeated multiplication. For large exponents it shows the logarithm method: bⁿ = e^(n × ln b). For fractional exponents it shows the radical form.

What is the difference between growth and decay in the table?

If the base is greater than 1, each period multiplies the value by a factor larger than 1 — growth. If the base is between 0 and 1, each period multiplies by a factor smaller than 1 — the value shrinks. A base of exactly 1 means no change; a base of 0 collapses to 0 after the first period.

Exponent Calculator

Inputs

Result

Settings

Growth Curve

Period Table

How to Use the Exponent Calculator

Exponent Formulas

Key Terms

Worked Examples

Exponents in the Real World

Finance & Compound Interest

Science & Nature

Computing & Information

Why Fractional Exponents Matter

Frequently Asked Questions

Exponent Calculator

Inputs

Result

Settings

Growth Curve

Period Table

How to Use the Exponent Calculator

Exponent Formulas

Key Terms

Worked Examples

Exponents in the Real World

Finance & Compound Interest

Science & Nature

Computing & Information

Why Fractional Exponents Matter

Frequently Asked Questions

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