Math · College algebra · Concept

Exponent rules and rational exponents

To multiply powers with the same base, add the exponents; to divide them, subtract; to raise a power to a power, multiply. A negative exponent means a reciprocal, x⁻ⁿ = 1/xⁿ, and a fractional exponent means a root: x^(m/n) is the nth root of xᵐ.

Multiplying and dividing powers

x³ · x² is (x · x · x)(x · x), five factors of x, so exponents add when powers with the same base are multiplied. Dividing cancels common factors, so exponents subtract.

x^{a} \cdot x^{b} = x^{a + b}, \frac{x^{a}}{x^{b}} = x^{a - b}

\begin{array}{c} x^{a} \cdot x^{b} = x^{a + b} \\ \frac{x^{a}}{x^{b}} = x^{a - b} \end{array}

Powers of powers and of products

Raising a power to a power multiplies the exponents. A power of a product or a quotient applies to every factor, including any number in front.

{(x^{a})}^{b} = x^{a b}, {(x y)}^{n} = x^{n} y^{n}

\begin{array}{c} {(x^{a})}^{b} = x^{a b} \\ {(x y)}^{n} = x^{n} y^{n} \end{array}

Zero and negative exponents

Since x³/x³ = 1 and the quotient rule gives x⁰, x⁰ = 1 for x ≠ 0. A negative exponent moves a factor across the fraction bar: x⁻² = 1/x², a positive number when x ≠ 0.

x^{0} = 1, x^{- n} = \frac{1}{x^{n}} (x \neq 0)

\begin{array}{c} x^{0} = 1, x^{- n} = \frac{1}{x^{n}} \\ (x \neq 0) \end{array}

Rational exponents are roots

x^(1/n) is the nth root of x, because its nth power is x. The numerator of a fractional exponent is a power and the denominator a root. Take the root first to keep the numbers small: 8^(2/3) = (∛8)² = 2² = 4.

x^{\frac{m}{n}} = {(\sqrt[n]{x})}^{m} = \sqrt[n]{x^{m}}

Simplifying radicals

Factor out the largest perfect square, then take its root: √72 = √(36 · 2) = 6√2. Radicals with the same radicand combine like terms, and multiplying top and bottom by √3 rationalizes 6/√3 = 2√3.

\sqrt{72} = \sqrt{36 \cdot 2} = 6 \sqrt{2}

Where the rules need care

The rules hold for positive bases with any real exponents. Even roots of negative numbers are not real, √(x²) = |x| rather than x, and removing a negative exponent can change where an expression is defined: x²/x² is undefined at 0, while 1 is not.

Common mistakes

Multiplying exponents when multiplying powers: x² · x³ = x⁵, not x⁶.
Distributing an exponent over a sum: (x + y)² is x² + 2xy + y², not x² + y².
Reading x⁻² as a negative number: it is 1/x², which is positive.
Raising only part of a product: (3x)² = 9x², while 3x² squares only x.

Key terms

Exponent: The raised number in a power, telling how many times to multiply the base: 2³ = 2·2·2. A negative exponent means a reciprocal (2⁻³ = 1/8), and a fractional one means a root.
Rational exponent: An exponent written as a fraction m/n, meaning the nth root raised to the mth power: x^(m/n) = (ⁿ√x)^m. For even n the base must be nonnegative to stay in the real numbers.
nth root: A number whose nth power is the given value. An odd root, such as ∛(−8) = −2, exists for every real number; in the real numbers an even root needs a nonnegative number.
Radical expression: An expression containing a root, such as √x or ∛(2x). The number under the radical sign is the radicand; simplifying pulls perfect powers out of it.
Exponential function: A function with the variable in the exponent, such as bˣ with a fixed base b > 0, b ≠ 1. The natural exponential eˣ is its own derivative.

Work through an example

Simplify (3x⁻²)² · x⁵ / (9x⁻³), and say where the result agrees with the original.

Simplify an expression with negative exponents →

Simplify and combine radicals →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

Make it concrete

Try in the workspace

Open the example inputs, change a value and keep a useful result on your board.

Check it for x > 0 in Steps, assumptions & inequalities Check each step in Math Open worked example on a board Exponent rules in Math Reference

Your existing work stays on this device. Examples open as editable copies.