Order of operations with powers and roots

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

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

Parentheses inside a root, power or fraction

Note that when there is, for example, an addition inside the radical symbol, this addition is actually inside implied parentheses. Therefore, we first calculate the sum inside the radical symbol.
This also applies to other operations inside the radical symbol.

The same applies to calculations in the numerator and the denominator of a fraction, and calculations inside an exponent: these calculations are actually inside parentheses.

Example

\[\begin{array}{rcl}\sqrt{3+6}&=& \sqrt{\left(3+6\right)}\\&=& \sqrt{9}\\&=& 3 \end{array}\]

Calculate #5 \times {3}^{3}#.

#5 \times {3}^{3}=# # 135#

#\begin{array}{rcl}
5 \times {3}^{3}&=&5 \times (3 \times 3 \times 3) \\
&&\phantom{xx}\blue{\text{exponentiation is repeated multiplication}}\\
&=&5 \times 27 \\
&&\phantom{xx}\blue{\text{calculated the expression within parentheses}}\\
&=& 135 \\
&&\phantom{xx}\blue{\text{multiplied}}\\
\end{array}#

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