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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 $263^2\times (86)$
For this example the order will be
 Brackets
 Squaring
 Multiplication
 Subtraction
$$ \begin{align*} 263^2\times {\color{red}{(86)}}&=26{\color{blue}{3^2}}\times {\color{red}{2}}\\ &=26{\color{blue}{9}}{\color{orange}{\times 2}}\\ &= 26{\color{orange}{18}}\\ &=8 \end{align*}$$
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 $22+\sqrt{259}+6$
Note carefully the difference between this and the previous example. Here the square root sign works as an implied bracket, that is $\sqrt{259}$ should be interpreted as $\sqrt{(259)}$. The order of operations will thus be the implied brackets, then taking the square root and finally addition from left to right: $$\begin{align*} 22+\sqrt{{\color{red}{259}}}+6&=22+{\color{blue}{\sqrt{{\color{red}{16}}}}}+6\\ &=22+{\color{blue}{4}}+6\\ &=32 \end{align*}$$
Try the following questions to confirm your understanding of these concepts.
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