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.

Exponent Calculator - Powers & Roots

Calculate x raised to the power n

raised to

Result

Scientific Notation

Expression

Calculate the nth root of x

th root of

Result

Scientific Notation

As Fractional Exponent

Find exponent: base^n = result

^ n =

Exponent (n)

Verification

Logarithm Form

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