Scientific Notation Calculator

Exponents Calculator

Use this exponents calculator to evaluate base–power expressions, apply the laws of exponents, and work with negative, zero, and fractional exponents instantly.

Evaluate base^exponent expressionsNegative and fractional exponentsLaws of exponents reference

Calculate powers, negative exponents, decimal exponents, and exponent expressions with clear steps.

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Base and exponent fields accept numbers, decimals, fractions, pi, e, sqrt, and parentheses.

Step by Step

Result Summary

Answer: 256
Scientific notation: 2.56 x 10^2
E notation: 2.56e2

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Exponents Calculator

An exponents calculator evaluates expressions involving a base raised to a power, including integer, negative, zero, and fractional exponents. Exponents appear in compound interest, population growth, radioactive decay, and nearly every branch of science and engineering. This calculator handles any valid base-exponent combination, applies the relevant law, and returns the result with a clear solution path.

What Are Exponents?

An exponent tells you how many times to multiply the base by itself. In 2³, the base is 2 and the exponent is 3, meaning 2 × 2 × 2 = 8. Exponents extend naturally to zero (any non-zero base raised to zero equals 1), to negative values (2⁻³ = 1/2³ = 1/8), and to fractions (8^(1/3) = ∛8 = 2). Fractional exponents combine powers and roots: a^(m/n) means the nth root of a raised to the mth power. Understanding these extensions lets you simplify complex expressions and solve equations across algebra, calculus, and applied mathematics.

How to Use the Exponents Calculator

Enter the base and the exponent into the input fields. The calculator accepts integers, decimals, and fractions for both values. Use the ^ symbol or the dedicated exponent button to enter powers. For compound expressions, use parentheses to group terms correctly before calculating.

Click Calculate. The results panel returns the evaluated numerical answer, the simplified form when applicable, and a step-by-step explanation showing which law of exponents was applied. For negative or fractional exponents, the calculator shows the intermediate reciprocal or root step explicitly.

Worked Examples

Representative examples: 2^10 = 1024; 5^0 = 1; 4^(−2) = 1/16 = 0.0625; 27^(1/3) = 3; 4^(3/2) = (√4)³ = 8; and (2^3)^2 = 2^6 = 64.

Common Exponent Problems

Frequently solved problems include evaluating large integer powers, simplifying expressions with negative exponents, applying the power-of-a-power rule, computing fractional exponents as roots, and verifying zero-exponent results for various bases.

Laws of Exponents Reference Tables

These tables list every major law of exponents with a worked example and the general algebraic form.

Product rule: multiply with the same base

When multiplying two expressions with the same base, add the exponents. The base stays the same.

Rule	General form	Example	Result
Product rule	a^m × a^n = a^(m+n)	2^3 × 2^4	2^7 = 128

Quotient rule: divide with the same base

When dividing two expressions with the same base, subtract the denominator exponent from the numerator exponent.

Rule	General form	Example	Result
Quotient rule	a^m / a^n = a^(m−n)	5^6 / 5^2	5^4 = 625

Power rule: raise a power to a power

When raising an exponent expression to another power, multiply the two exponents together.

Rule	General form	Example	Result
Power rule	(a^m)^n = a^(m×n)	(3^2)^4	3^8 = 6561

Zero exponent rule

Any non-zero base raised to the power of zero equals exactly 1. This follows directly from the quotient rule applied to a^n / a^n.

Rule	General form	Example	Result
Zero exponent	a^0 = 1 (a ≠ 0)	7^0	1
Any non-zero base	a^0 = 1	(−5)^0	1

Negative exponent rule

A negative exponent indicates a reciprocal. The expression a^(−n) equals 1 / a^n. This produces a fraction, not a negative number.

Rule	General form	Example	Result
Negative exponent	a^(−n) = 1 / a^n	2^(−3)	1/8 = 0.125
Fraction base	(1/3)^(−2)	(3)^2	9

Fractional exponent rule

A fractional exponent a^(m/n) equals the nth root of a raised to the mth power. The denominator is the root index and the numerator is the power.

Rule	General form	Example	Result
Fractional exponent	a^(m/n) = (ⁿ√a)^m	8^(2/3)	(∛8)² = 4
Square root form	a^(1/2) = √a	25^(1/2)	5

Power distribution to a product or quotient

When the same exponent applies to a product or quotient, distribute it to each factor separately.

Rule	General form	Example	Result
Product power	(a × b)^n = a^n × b^n	(2 × 3)^4	16 × 81 = 1296
Quotient power	(a/b)^n = a^n / b^n	(4/5)^2	16/25 = 0.64

Process

Calculation Workflow

Enter base and exponent

Identify applicable law

Evaluate expression

Return simplified result

The calculator reads the base and exponent, identifies which law of exponents applies, performs the calculation step by step, and returns the numeric result alongside the simplified form and the rule that was used.

Input Formats

Supported Formats

Input style	Example
Integer base and exponent	2^10
Negative exponent	5^(−3)
Fractional exponent	27^(1/3)
Power of a power	(3^2)^4

Reference

Common Scientific Constants

Back to calculator

Constant name	Symbol	Notation value	Unit
Speed of light	c	2.99792458 x 10^8	m/s
Gravitational constant	G	6.67430 x 10^-11	N m^2/kg^2
Avogadro constant	NA	6.02214076 x 10^23	mol^-1

Questions

Calculator FAQ

What is an exponent?

An exponent indicates how many times a base is multiplied by itself. In 3^4, the base is 3 and the exponent is 4, meaning 3 × 3 × 3 × 3 = 81.

What does a negative exponent mean?

A negative exponent means take the reciprocal of the base raised to the positive version of that exponent. For example, 5^(−2) = 1 / 5^2 = 1/25 = 0.04.

What is anything to the power of zero?

Any non-zero number raised to the power of zero equals 1. This is the zero exponent rule and holds for all non-zero bases.

What does a fractional exponent mean?

A fractional exponent combines a root and a power. The denominator is the root index and the numerator is the power. For example, 8^(2/3) = (∛8)² = 2² = 4.

What is the product rule for exponents?

When multiplying two powers with the same base, add the exponents: a^m × a^n = a^(m+n). For example, 3^2 × 3^5 = 3^7 = 2187.

What is the quotient rule for exponents?

When dividing two powers with the same base, subtract the denominator exponent from the numerator: a^m / a^n = a^(m−n). For example, 5^6 / 5^2 = 5^4 = 625.

What is the power of a power rule?

When raising an exponential expression to another exponent, multiply the two exponents: (a^m)^n = a^(m×n). For example, (2^3)^4 = 2^12 = 4096.

How do exponents relate to roots?

Fractional exponents and roots are two notations for the same operation. The exponent 1/n is the nth root: a^(1/2) = √a, a^(1/3) = ∛a, and so on.

How do I simplify 4^(3/2)?

Rewrite as (√4)^3. √4 = 2, then 2^3 = 8. Alternatively, 4^3 = 64 first, then √64 = 8. Both approaches give 4^(3/2) = 8.

Can the base be negative?

Yes. A negative base raised to an even exponent gives a positive result, and an odd exponent gives a negative result. For example, (−2)^4 = 16 and (−2)^3 = −8.

What is 0^0?

Zero raised to zero is a special case. In most mathematical contexts it is defined as 1, particularly in combinatorics and calculus, but the expression is technically considered indeterminate.

What is the rule for a power applied to a product?

When an exponent applies to a product of two numbers, distribute it to each factor: (a × b)^n = a^n × b^n. For example, (2 × 5)^3 = 2^3 × 5^3 = 1000.

What is an exponential function?

An exponential function has the form f(x) = a^x, where a is the base and x is the variable exponent. These functions model compound interest, population growth, and radioactive decay.

How do exponents apply to unit conversions?

When converting squared or cubed units, apply the exponent to the conversion factor. For example, 1 m² = 10,000 cm² because 1 m = 100 cm and (100)^2 = 10,000.

How do I calculate a number raised to a large exponent?

Enter the base and exponent into the calculator. For very large results, the output uses scientific notation. You can also use logarithms: log(a^n) = n × log(a).