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Exponent Calculator

Enter a base and exponent to calculate the result. Works with positive, negative, zero, and fractional exponents. Step-by-step work is shown for each calculation.

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Results

2^10 =1,024
Scientific notation:1.0240e+3
Step-by-step work:
2^10 = 2 multiplied by itself 10 times

  2^10 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
  2^10 = 1,024

What Is an Exponent?

An exponent indicates how many times a number (the base) is multiplied by itself. The expression bn is read "b to the n-th power" or "b raised to n." The base b is the number being multiplied, and the exponent n (also called the power or index) tells you how many times.

Examples: 53 = 5 × 5 × 5 = 125.   106 = 1,000,000.   210 = 1,024.

Exponents are a compact way to write repeated multiplication. They appear in geometry (area = side², volume = side³), science (powers of 10 in scientific notation), finance (compound interest = (1 + r)n), and computer science (binary uses powers of 2).

Laws of Exponents

The laws of exponents let you simplify expressions without computing each power individually. All of these rules follow from the basic definition of exponentiation as repeated multiplication.

Rule name Formula Example
Product rule bm × bn = bm+n 23 × 24 = 27 = 128
Quotient rule bm / bn = bm−n 35 / 32 = 33 = 27
Power rule (bm)n = bmn (23)4 = 212 = 4096
Product base (ab)n = an × bn (2 × 3)4 = 24 × 34 = 1296
Zero exponent b0 = 1 (b ≠ 0) 70 = 1
Negative exponent b−n = 1 / bn 2−3 = 1/8 = 0.125
Fractional exponent b1/n = n√b 81/3 = ³√8 = 2

Negative Exponents Explained

A negative exponent does not make the result negative — it means take the reciprocal of the positive power. This can be understood from the quotient rule: if you divide b2 by b5, you get b−3, which must equal 1/b3 because dividing a smaller power by a larger one gives a fraction.

Example: 5−2 = 1 / 52 = 1/25 = 0.04

This is the basis for scientific notation: 3 × 10−4 = 3 / 10,000 = 0.0003.

Fractional Exponents and Roots

A fractional exponent bm/n combines a power and a root. The denominator n gives the root index, and the numerator m gives the power:

bm/n = (n√b)m = n√(bm)

Both orders (root then power, or power then root) give the same result.

Examples:
641/2 = √64 = 8
272/3 = (³√27)2 = 32 = 9
163/4 = (4√16)3 = 23 = 8

Scientific Notation and Powers of 10

Scientific notation uses exponents to represent very large or very small numbers concisely. A number in scientific notation has the form a × 10n, where 1 ≤ |a| < 10.

Standard form Scientific notation Power of 10
1,000,0001 × 106106
300,0003 × 105105
0.0011 × 10−310−3
0.0000454.5 × 10−510−5

Frequently Asked Questions

What is an exponent?

An exponent (also called a power) tells you how many times to multiply a number (the base) by itself. In the expression bⁿ, b is the base and n is the exponent. For example, 2³ = 2 × 2 × 2 = 8. Exponents are a compact notation for repeated multiplication. The expression is read "b to the power of n" or "b to the n-th power."

What is the rule for a negative exponent?

A negative exponent means the reciprocal of the positive exponent: b⁻ⁿ = 1 / bⁿ. For example, 2⁻³ = 1 / 2³ = 1/8 = 0.125. This rule follows from the quotient rule: bⁿ / bᵐ = bⁿ⁻ᵐ. If n = 0 and m = 3, you get b⁰ / b³ = 1 / b³ = b⁻³. Negative exponents never make a result negative — they make it a fraction between 0 and 1 (for a positive base greater than 1).

What does a fractional exponent mean?

A fractional exponent represents a root: b^(1/n) = the n-th root of b, and b^(m/n) = the n-th root of bᵐ (or equivalently, (n-th root of b)ᵐ). For example, 8^(1/3) = ∛8 = 2, and 27^(2/3) = (∛27)² = 3² = 9. The denominator of the fraction is the root index and the numerator is the power. This unifies the concepts of powers and roots into a single notation.

What is anything raised to the power of zero?

Any non-zero number raised to the power of 0 equals 1: b⁰ = 1 (for b ≠ 0). This follows from the quotient rule: bⁿ / bⁿ = bⁿ⁻ⁿ = b⁰, and any number divided by itself equals 1. The expression 0⁰ is mathematically indeterminate — different branches of mathematics define it differently, though in combinatorics and analysis it is often taken to be 1 as a convention.

What are the laws of exponents?

The main laws of exponents are: (1) Product rule: bᵐ × bⁿ = bᵐ⁺ⁿ. (2) Quotient rule: bᵐ / bⁿ = bᵐ⁻ⁿ. (3) Power rule: (bᵐ)ⁿ = bᵐⁿ. (4) Product base rule: (ab)ⁿ = aⁿ × bⁿ. (5) Quotient base rule: (a/b)ⁿ = aⁿ/bⁿ. (6) Zero exponent: b⁰ = 1. (7) Negative exponent: b⁻ⁿ = 1/bⁿ. (8) Fractional exponent: b^(m/n) = (n-th root of b)ᵐ. These rules let you simplify complex expressions without evaluating them numerically.

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