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# Laws of indices

There are three laws of indices. They relate operations on exponents with the same base to operations on the indices.

• Addition of indices: When the exponents are multiplied, the powers are added. $$\Tblue{3}^\Tred{2} \times \Tblue{3}^\Torange{5} = (\Tblue{3} \times \Tblue{3}) \times (\Tblue{3} \times \Tblue{3} \times \Tblue{3} \times \Tblue{3} \times \Tblue{3}) = \Tblue{3}^7 = \Tblue{3}^{\Tred{2}+\Torange{5}}$$
• Subtraction of indices: When the exponents are divided, the powers are subtracted. \begin{align*} \Tblue{2}^\Tred{4} \div \Tblue{2}^\Torange{2} &= (\Tblue{2} \times \Tblue{2} \times \Tblue{2} \times \Tblue{2}) \div (\Tblue{2} \times \Tblue{2}) \\&= \frac{\Tblue{2} \times \Tblue{2} \times \Tblue{2} \times \Tblue{2}}{\Tblue{2} \times \Tblue{2}} = \Tblue{2}^2 = \Tblue{2}^{\Tred{4}-\Torange{2}} \end{align*}
• Multiplication of indices: When an exponent is raised to a power, the powers are multiplied. $$(\Tblue{3}^\Tred{2})^\Torange{3} = (\Tblue{3} \times \Tblue{3}) \times (\Tblue{3} \times \Tblue{3}) \times (\Tblue{3} \times \Tblue{3}) = \Tblue{3}^6 = \Tblue{3}^{\Tred{2} \times\Torange{3}}$$
Index Exponent Formula Example
$+$ $\times$ $\Tblue{a}^{\Tred{x}+\Torange{y}}=\Tblue{a}^\Tred{x}\Tblue{a}^\Torange{y}$ $\Tblue{2}^{\Tred{2}+\Torange{1}} = \Tblue{2}^\Tred{2}\cdot \Tblue{2}^\Torange{1} = \Tblue{2}^3$
$-$ $\div$ $\displaystyle \Tblue{a}^{\Tred{x}-\Torange{y}}=\frac{\Tblue{a}^\Tred{x}}{\Tblue{a}^\Torange{y}}$ $\displaystyle \Tblue{2}^{\Tred{5}-\Torange{2}} = \frac{\Tblue{2}^\Tred{5}}{\Tblue{2}^\Torange{2}} = \Tblue{2}^3$
$\times$ Power $(\Tblue{a}^{\Tred{x}})^{\Torange{y}}=\Tblue{a}^{\Tred{x}\Torange{y}}$ $(\Tblue{2}^\Tred{2})^\Torange{2}= \Tblue{2}^{\Tred{2}\times \Torange{2}} = \Tblue{2}^4$