Multiplication of powers

When we multiply #\blue2^\orange3# by #\blue2^\purple4#, we get:

\[\begin{array}{rcl}\blue2^\orange3 \times \blue2^\purple4&=&\underbrace{(\blue2 \times \blue2 \times \blue2)} _{\orange3 \text { times}}\times \underbrace{(\blue2 \times \blue2 \times \blue2 \times \blue2)}_{\purple4 \text{ times}}\\&=&\blue2^{\orange3+\purple4}\\&=&\blue2^7\end{array}\]

In general, we can state the following:

When multiplying powers with the same #\blue{\textit{base}}#, we should add the exponents. The #\blue{\textit{base}}# remains the same.

Examples

\[\begin{array}{rcl}\blue3^\orange{4} \times \blue3^\purple{2} &=&\blue3^{\orange4+\purple2} \\&=&\blue3^{6} \\ \\ (\blue{\frac{1}{2}})^\orange{5} \times (\blue{\frac{1}{2}})^\purple3&=&(\blue{\frac{1}{2}})^{\orange5+\purple3} \\&=&(\blue{\frac{1}{2}})^{8} \\ \\ (\blue{-3})^\orange{10} \times (\blue{-3})^\purple{5}&=&(\blue{-3})^{\orange{10}+\purple5} \\&=&(\blue{-3})^{15} \end{array}\]

Division of powers

When we divide #\blue2^\orange{5}# by #\blue2^\purple2#, we get:

\[\begin{array}{rcl}\blue2^\orange5 \div \blue2^\purple2&=&\underbrace{(\blue2 \times \blue2 \times \blue2 \times \blue2 \times \blue2)}_{\orange5 \text{ times}} \div \underbrace{(\blue2 \times \blue2)}_{\purple2 \text{ times}}\\&=&\underbrace{\blue2 \times \blue2 \times \blue2}_{\orange5-\purple2 \text { times}}\\&=&\blue2^{\orange5-\purple2}\\&=&\blue2^3\end{array}\]

In general, we can state the following:

When dividing powers with the same #\blue{\textit{base}}#, we should subtract the exponents. The #\blue{\textit{base}}# remains the same.

Examples

\[\begin{array}{rcl}\blue3^\orange{4} \div \blue3^\purple{2}&=&\blue3^{\orange4-\purple2} \\&=&\blue3^{2} \\ \\ (\blue{\frac{1}{2}})^\orange{5} \div (\blue{\frac{1}{2}})^\purple3&=&(\blue{\frac{1}{2}})^{\orange5-\purple3} \\&=&(\blue{\frac{1}{2}})^{2} \\ \\ (\blue{-3})^\orange{10} \div (\blue{-3})^\purple{5}&=&(\blue{-3})^{\orange{10}-\purple5} \\&=&(\blue{-3})^{5}\end{array}\]

Exponentiation of powers

When we raise #\blue2^\orange{3}# to the power #\purple4#, we get:

\[\begin{array}{rcl}\left(\blue2^\orange3\right)^\purple4&=&\underbrace{\blue2^\orange3 \times \blue2^\orange3 \times \blue2^\orange3 \times \blue2^\orange3}_{\purple4 \text { times}}\\&=&\blue2^{\overbrace{\orange3+\orange3+\orange3+\orange3}^{\purple4 \text{ times}}}\\&=&\blue2^{\orange3 \times \purple4}\\&=&\blue2^{12}\end{array}\]

In general, we can state the following:

When raising a power to a power, we should multiply the exponents. The #\blue{\textit{base}}# remains the same.

Example

\[\begin{array}{rcl}\left(\blue3^\orange{4}\right)^\purple{2} &=& \blue3^{\orange4 \times \purple2} \\ &=& \blue3^{8} \\ \\ \left(\left(\blue{\frac{1}{2}}\right)^\orange{5}\right)^\purple3&=&(\blue{\frac{1}{2}})^{\orange5 \times \purple3} \\&=&(\blue{\frac{1}{2}})^{15} \\ \\ \left(\left(\blue{-3}\right)^\orange{10}\right)^\purple{5} &=& (\blue{-3})^{\orange{10} \times \purple5} \\&=&(\blue{-3})^{50} \end{array}\]