Laws of Indices

Law 1: Multiplication Law

When you multiply similar terms, you need to add their powers.

a^{\textcolor{blue}m} \times a^{\textcolor{red}n}  = a^{\textcolor{blue}m+\textcolor{red}n} 

The multiplication law applies to all numbers, negative numbers and fractional powers.

Example:

\begin{aligned} a^3 a^4 &= a^{3+4} = a^7 \\ x^4 x^{-1} &= x^{4-1} = x^3 \\ (x+1)^2 (x+1)^3 &= (x+1)^{2+3} = (x+1)^5 \\ t^{\frac{1}{5}} t^{\frac{2}{5}} &= t^{ \frac{1}{5} + \frac{2}{5}} = t^{\frac{3}{5}} \\ xy^2 \cdot x^3 y^{-1} &= x^{1+3} y^{2-1} = x^4 y \end{aligned}

Note: When there are multiple variables, you need to add the powers separately for each variable.

Law 2: Division Law

When you divide similar terms, you need to subtract their powers.

\dfrac{a^{\textcolor{blue}m}}{a^{\textcolor{red}n}} = a^{\textcolor{blue}m} \div a^{\textcolor{red}n}  = a^{\textcolor{blue}m - \textcolor{red}n} 

The division law applies to all numbers, negative numbers and fractional powers.

Example:

\begin{aligned} \dfrac{a^6}{a^4} &= a^{6-4} = a^2 \\[1.2em] \dfrac{x^3}{x^{-1}} &= x^{3-(-1)} = x^4 \\[1.2em] \dfrac{y^2}{y^\frac{1}{2}} &= y^{2 - \frac{1}{2}} = y^{\frac{3}{2}} \\[1.2em] \dfrac{x^2 y^4}{x^3 y} &= x^{2-3} y^{4-1} = x^{-1} y^3 \end{aligned}

Note: When there are multiple variables, you need to subtract the powers separately for each variable.

Law 3: Multiple Powers Law

If you have a power that is raised to another power, then you multiply the powers.

(a^{\textcolor{blue}m})^\textcolor{red}n = a^{\textcolor{blue}m \textcolor{red}n} 

The multiple powers law applies to all numbers, negative numbers and fractional powers.

Example:

\begin{aligned} (x^3)^2 &= x^{3 \times 2} = x^6 \\ (y^4)^{-2} &= y^{4 \times -2} = y^{-8} \\ (x^2y)^3 &= (x^2)^3 y^{1 \times 3} = x^{2 \times 3} y^3 = x^6 y^3 \end{aligned}

Note: When there are multiple variables, you need to multiply the powers separately for each variable.

Law 4: Power 0 Law

Any number or letter to the power 0 = 1

a^{\textcolor{red}0} = \textcolor{blue}1

Example:

\begin{aligned} 12^0 &= 1 \\ x^0 &= 1 \end{aligned}

Law 5: Roots as Powers Law

Roots, for example square roots or cube roots, can be written as powers.

a^{\frac{1}{\textcolor{blue}m}} = \sqrt[\textcolor{blue}m]{a}

Example:

\begin{aligned} 9^{\frac{1}{2}} &= \sqrt{9} = 3 \\ 64^{\frac{1}{3}} &= \sqrt[3]{64} = 4 \\ x^{\frac{1}{4}} &= \sqrt[4]{x} \end{aligned}

Law 6: Fractional Powers Law

A power that is represented as a fraction means the power of a root or the root of a power. This extends on law 5.

{a}^{{\frac{\textcolor{blue}{m}}{\textcolor{red}{n}}}} = \sqrt[\textcolor{red}{n}]{{a}^\textcolor{blue}{m}} =(\sqrt[\textcolor{red}{n}]{{a}})^\textcolor{blue}{m}

Example:

\begin{aligned} 16^{\frac{3}{2}} &= (16^{\frac{1}{2}})^3 = (\sqrt{16})^3 = 4^3 = 64 \\ 27^{\frac{2}{3}} &= (27^{\frac{1}{3}})^2 = (\sqrt[3]{27})^2 = 3^2 = 9 \end{aligned}

Law 7: Negative Powers Law

A negative power puts the positive power on the bottom of a fraction.

a^{-\textcolor{blue}m} = \dfrac{1}{a^{\textcolor{blue}m}}

Example:

\begin{aligned} 3^{-2} &= \dfrac{1}{3^2} = \dfrac{1}{9} \\[1.2em] (3x-1)^{-1} &= \dfrac{1}{3x-1} \end{aligned}