### Numbers: Powers and roots

### Order of operations with powers and roots

We have already seen the order of operations for addition, subtraction, multiplication and division. Now we will look at where powers and roots fit in this order.

Order of operations

When we have an expression that contains multiple operations, we adhere to the following order.

- Perform calculations that are between parentheses.
- Calculate powers and roots.
- Multiply and divide from left to right.
- Add and subtract from left to right.

**Example**

\[\begin{array}{rcl}\sqrt{16}-\left(2+5\right) \times 2^2 &=& \sqrt{16}-7\times 2^2\\&=&4-7\times 4 \\&=&4-28 \\ &=& -24 \end{array}\]

#5 \times {2}^{3}=# # 40#

#\begin{array}{rcl}

5 \times {2}^{3}&=&5 \times (2 \times 2 \times 2) \\

&&\phantom{xx}\blue{\text{exponentiation is repeated multiplication}}\\

&=&5 \times 8 \\

&&\phantom{xx}\blue{\text{calculated the expression within parentheses}}\\

&=& 40 \\

&&\phantom{xx}\blue{\text{multiplied}}\\

\end{array}#

#\begin{array}{rcl}

5 \times {2}^{3}&=&5 \times (2 \times 2 \times 2) \\

&&\phantom{xx}\blue{\text{exponentiation is repeated multiplication}}\\

&=&5 \times 8 \\

&&\phantom{xx}\blue{\text{calculated the expression within parentheses}}\\

&=& 40 \\

&&\phantom{xx}\blue{\text{multiplied}}\\

\end{array}#

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