Algebra visual reference

Exponent Rules Chart

Exponent rules describe repeated multiplication. They apply only when the bases and operations match the rule, so the chart pairs every symbolic pattern with a short example.

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$Exponent rules chart showing product quotient power zero negative and fractional exponent laws with worked examples$

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How to read it

Use the chart as a decision tool.

A printable exponent rules chart with symbolic laws and quick examples for products, quotients, powers, zero, negative, and fractional exponents. Read the label first, identify the matching structure in your problem, then confirm that any stated conditions apply before substituting values.

Combine powers

These rules require a shared base.

Product rule

\displaystyle a^m a^n=a^{m+n}

Example: $\displaystyle x^3x^5=x^8$

Quotient rule

\displaystyle \frac{a^m}{a^n}=a^{m-n}

Example: $\displaystyle \frac{y^7}{y^2}=y^5$

Powers of powers and products

Distribute the outside exponent to every factor.

Power rule

\displaystyle (a^m)^n=a^{mn}

Example: $\displaystyle (x^2)^3=x^6$

Power of a product

\displaystyle (ab)^n=a^n b^n

Example: $\displaystyle (2x)^3=8x^3$

Power of a quotient

\displaystyle \left(\frac ab\right)^n=\frac{a^n}{b^n}

Example: $\displaystyle \left(\frac x3\right)^2=\frac{x^2}{9}$

Special exponents

Assume the base is nonzero where division occurs.

Zero exponent

\displaystyle a^0=1

Example: $\displaystyle 7^0=1$

Negative exponent

\displaystyle a^{-n}=\frac1{a^n}

Example: $\displaystyle x^{-3}=\frac1{x^3}$

Fractional exponent

\displaystyle a^{1/n}=\sqrt[n]{a}

Example: $\displaystyle 27^{1/3}=3$

How to remember it

Decide what operation connects the powers

1Same base multiplied: add exponents.
2Same base divided: subtract exponents.
3A power raised to a power: multiply exponents.
4A negative exponent means reciprocal, not a negative value.