Order of operations

Algebra: Calculating with exponents and roots

Order of operations

Now that we have explained all operations, we will check in which order we are allowed to perform them.

When adding or subtraction, we perform the operations from left to right.

Example
\begin{array}{rcl}
\blue{3x+x} - 2x &=& \blue{4x} - 2x \\
&=& 2x
\end{array}

When multiplying or dividing, we perform the operations from left to right.

Example
\begin{array}{rcl}
\blue{4x : 2x} \cdot 3x &=& \blue{2} \cdot 3x \\
&=& 6x
\end{array}

Multiplying or dividing has precedence over adding and subtracting.

Example
\begin{array}{rcl}
2x+\blue{2\cdot 4 x} &=& 2x+\blue{8x} \\
&=& 10x
\end{array}

Exponentiation has precedence over multiplying and dividing.

Example
\begin{array}{rcl}
x\cdot \blue{2^3} &=& x\cdot \blue{8} \\
&=& 8x
\end{array}

First, calculate what is in between brackets.

Example
\begin{array}{rcl}
\blue{(3x+5x)}\cdot 8 &=& \blue{8x}\cdot {8} \\
&=& 64x
\end{array}

Simplify #4 \cdot x^{3+2} \cdot x + 4 \cdot x^{3+1} \cdot 6 \cdot x^{2}#.

#28 \cdot x^{6}#

#\begin{array}{rcl} 4 \cdot x^{3+2} \cdot x + 4 \cdot x^{3+1} \cdot 6 \cdot x^{2} &=& 4 \cdot x^{5} \cdot x + 4 \cdot x^{4} \cdot 6 \cdot x^{2} \\
&& \phantom{xxx}\blue{\text{exponentiation has precedence }}\\
&=& 4 \cdot x^{6} + 24 \cdot x^{6} \\
&& \phantom{xxx}\blue{\text{multiplication is the second step }} \\ && \phantom{xxx}\blue{\text{with the rule for exponents: }a^{n} \cdot a^{m}=a^{n+m}}\\
&=& 28 \cdot x^{6} \\
&& \phantom{xxx}\blue{\text{then adding/subtracting like terms }}\\
\end{array}#

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