Exponent Calculator

Calculate powers, roots, and find exponents with step-by-step solutions. Supports negative exponents, fractional exponents, and scientific notation.

Calculate x raised to the power n

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Scientific Notation
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Expression
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Calculate the nth root of x

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Scientific Notation
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As Fractional Exponent
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Find exponent: base^n = result

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Verification
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Logarithm Form
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Laws of Exponents

Product Rule
x^a * x^b = x^(a+b)
Example: 2^3 * 2^4 = 2^7 = 128
Quotient Rule
x^a / x^b = x^(a-b)
Example: 2^5 / 2^2 = 2^3 = 8
Power Rule
(x^a)^b = x^(a*b)
Example: (2^3)^2 = 2^6 = 64
Negative Exponent
x^(-n) = 1 / x^n
Example: 2^(-3) = 1/8 = 0.125
Fractional Exponent
x^(m/n) = n-th root of x^m
Example: 8^(2/3) = (cbrt 8)^2 = 4
Product to Power
(xy)^n = x^n * y^n
Example: (2*3)^2 = 4 * 9 = 36

Special Exponent Cases

Rule Formula Example
Zero Exponent x^0 = 1 5^0 = 1, 100^0 = 1
Exponent of One x^1 = x 7^1 = 7, 42^1 = 42
Base of One 1^n = 1 1^5 = 1, 1^100 = 1
Base of Zero 0^n = 0 (n > 0) 0^5 = 0, 0^100 = 0
Negative Base (even exp) (-x)^even = positive (-2)^4 = 16
Negative Base (odd exp) (-x)^odd = negative (-2)^3 = -8

Frequently Asked Questions

What does a negative exponent mean?

A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, x^(-n) = 1/x^n. So 2^(-3) = 1/2^3 = 1/8 = 0.125. This rule applies to any non-zero base.

How do you calculate a fractional exponent?

A fractional exponent like x^(m/n) means taking the nth root of x raised to the power m. For example, 8^(2/3) = (cube root of 8)^2 = 2^2 = 4. The denominator is the root and the numerator is the power.

Why does any number raised to the power of 0 equal 1?

Any non-zero number raised to the power of 0 equals 1 because of the pattern of exponents. When you divide x^n by x^n, you get x^(n-n) = x^0, and any number divided by itself equals 1. This maintains consistency in exponential rules.

How do you find what exponent gives a certain result?

To find the exponent n where base^n = result, use logarithms: n = log(result) / log(base). For example, to find n where 2^n = 8, calculate n = log(8) / log(2) = 3. This is the inverse operation of exponentiation.