# Operations with Powers

After completing any expressions in brackets, the next step is to calculate any numbers that are raised to a power or where a square root is required. When there is an operation under the square root sign, this must be done first as the square root sign functions as brackets.

Example 1      $2\times 5^2$

Here we square the $5$ first then complete the multiplication: \begin{align*} 2\times {\color{red}{5^2}}&=2\times {\color{red}{25}}\\ &=50 \end{align*}

Example 2      $26-3^2\times (8-6)$

For this example the order will be

1. Brackets
2. Squaring
3. Multiplication
4. Subtraction
\begin{align*} 26-3^2\times {\color{red}{(8-6)}}&=26-{\color{blue}{3^2}}\times {\color{red}{2}}\\ &=26-{\color{blue}{9}}{\color{orange}{\times 2}}\\ &= 26-{\color{orange}{18}}\\ &=8 \end{align*}

Example 3      $22-\left( 4\times -1\right)^2$

Here we will simplify the bracket, then square the result and finally complete the subtraction: \begin{align*} 22-{\color{red}{\left( 4\times -1\right)}}^2&=22-{\color{red}{\left( -4\right)}}^{\color{blue}{2}}\\ &=22-{\color{blue}{16}}\\ &=6 \end{align*}

Example 4      $22+\sqrt{25}-\sqrt{9}+6$

This time we first extract the square roots, then carry out the addition and subtraction from left to right: \begin{align*} 22+{\color{red}{\sqrt{25}}}-{\color{red}{\sqrt{9}}}+6&=22+{\color{red}{5}}-{\color{red}{3}}+6\\ &=30 \end{align*}

Example 5      $22+\sqrt{25-9}+6$

Note carefully the difference between this and the previous example. Here the square root sign works as an implied bracket, that is $\sqrt{25-9}$ should be interpreted as $\sqrt{(25-9)}$. The order of operations will thus be the implied brackets, then taking the square root and finally addition from left to right: \begin{align*} 22+\sqrt{{\color{red}{25-9}}}+6&=22+{\color{blue}{\sqrt{{\color{red}{16}}}}}+6\\ &=22+{\color{blue}{4}}+6\\ &=32 \end{align*}