Let's Review
To simplify an order-of-operations expression, you need to start with the parentheses. Sometimes brackets, braces, or absolute value symbols are also used as grouping symbols, and sometimes these grouping symbols are nested within each other. When this happens, you need to find the innermost set of grouping symbols and work your way out, applying the order of operations as you go.
Keep in Mind
When math problems become more complicated, it is sometimes necessary to have groups inside groups. Grouping symbols inside other grouping symbols is called "nested grouping symbols." For nested grouping symbols, always work from the inside out.
For Example
Simplify \( - 1 \cdot \left| 2^{3} + \left( 3 + 4 \right)^{2} \right| \)
This expression contains multiple grouping symbols. Find the innermost set and begin working your way out. |
\( - 1 \cdot \left| 2^{3} + \left( \require{color}\colorbox{yellow}{$ 3 + 4 $} \right)^{2} \right| \) \( - 1 \cdot \left| 2^{3} + \left( \require{color}\colorbox{yellow}{$ 7 $} \right)^{2} \right| \) |
Simplify the exponential notation. |
\( - 1 \cdot \left| \require{color}\colorbox{yellow}{$ 2^{3} $} + \left( 7 \right)^{2} \right| \) \( - 1 \cdot \left| \require{color}\colorbox{yellow}{$ 8 $} + \require{color}\colorbox{lime}{$ \left( 7 \right)^{2} $} \right| \) \( - 1 \cdot \left| 8 + \require{color}\colorbox{lime}{$ 49 $} \right| \) |
Simplify the addition inside the absolute value symbols. |
\( - 1 \cdot \left| \require{color}\colorbox{yellow}{$ 8 + 49 $} \right| \) \( - 1 \cdot \left| \require{color}\colorbox{yellow}{$ 57 $} \right| \) |
Find the absolute value of 57. |
\( - 1 \cdot 57 \) |
Multiply. |
\( - 1 \cdot 57 \) \( - 57 \) |
Think you got it?
How well can you simplify larger order-of-operations expressions? Use the activity below to find out. Simplify the expression on each tab. Then check your answer.
Simplify \( 15 - \left| - 2 - 3^{2} \right| + 12 \)
16
If you need help arriving at this answer, click the solution button.
Work within the absolute value symbols first. |
\( 15 - \left| - 2 - \require{color}\colorbox{yellow}{$ 3^{2} $} \right| + 12 \) \( 15 - \left| \require{color}\colorbox{lime}{$ - 2 - $} \require{color}\colorbox{yellow}{$ 9 $} \right| + 12 \) \( 15 - \left| \require{color}\colorbox{lime}{$ - 11 $} \right| + 12\ \) \( 15 - \require{color}\colorbox{aqua}{$ 11 $} + 12 \) |
Add and subtract from left to right. |
\( 15 - 11 + 12 \) \( 4 + 12 \) \( 16 \) |
Simplify \( (6 \div 12) \cdot \lbrack(84 - 9) \div 5 + 1\rbrack \)
8
If you need help arriving at this answer, click the solution button.
Start within the innermost set of grouping symbols. |
\( (6 \div 12) \cdot \lbrack(\require{color}\colorbox{yellow}{$ 84 - 9 $}) \div 5 + 1\rbrack \) \( (6 \div 12) \cdot \lbrack \require{color}\colorbox{yellow}{$ 75 $} \div 5 + 1\rbrack \) |
Simplify within the brackets, using the order of operations. |
\( (6 \div 12) \cdot \lbrack \require{color}\colorbox{yellow}{$ 75 \div 5 $} + 1\rbrack \) \( (6 \div 12) \cdot \lbrack \require{color}\colorbox{yellow}{$ 15 $} \require{color}\colorbox{lime}{$ + 1 $}\rbrack \) \( (6 \div 12) \cdot \require{color}\colorbox{lime}{$ 16 $} \) |
Simplify within the parentheses. |
\( \left( \require{color}\colorbox{yellow}{$ 6 \div 12 $} \right) \cdot 16 \) \( \require{color}\colorbox{yellow}{$ \frac{1}{2} $} \cdot 16 \) \( 8 \) |
Simplify \( \left| - 1 - 4 \cdot \left( 50 - 8^{2} \right) \right| \div 11 \)
5
If you need help arriving at this answer, click the solution button.
Start within the parentheses. Remember that a negative number multiplied by another negative number results in a positive product. |
\( \left| - 1 - 4 \cdot \left( 50 - \require{color}\colorbox{yellow}{$ 8^{2} $} \right) \right| \div 11 \) \( \left| - 1 - 4 \cdot \left( \require{color}\colorbox{lime}{$ 50 - $} \require{color}\colorbox{yellow}{$ 64 $} \right) \right| \div 11 \) \( \left| - 1 - \require{color}\colorbox{aqua}{$ 4 \cdot $} \left( \require{color}\colorbox{lime}{$ - 14 $} \right) \right| \div 11 \) \( \left| \require{color}\colorbox{violet}{$ - 1 + $} \require{color}\colorbox{aqua}{$ 56 $} \right| \div 11 \) \( \left| \require{color}\colorbox{violet}{$ 55 $} \right| \div 11 \) \( 55 \div 11 \) |
Divide. |
\( 55 \div 11 \) \( 5 \) |
Simplify \( 5 \cdot \left( 16 - 2^{4} \right) + 8 \cdot (4 - \left| 12^{2} - 11 \cdot 12 \right|) \)
\( - 64 \)
If you need help arriving at this answer, click the solution button.
Simplify within the absolute value symbols and work your way out. Be sure to follow the order of operations along the way. |
\( 5 \cdot \left( 16 - 2^{4} \right) + 8 \cdot (4 - \left| \require{color}\colorbox{yellow}{$ 12^{2} $} - 11 \cdot 12 \right|) \) \( 5 \cdot \left( 16 - 2^{4} \right) + 8 \cdot (4 - \left| \require{color}\colorbox{yellow}{$ 144 $} - \require{color}\colorbox{lime}{$ 11 \cdot 12 $} \right|) \) \( 5 \cdot \left( 16 - 2^{4} \right) + 8 \cdot (4 - \left| \require{color}\colorbox{aqua}{$ 144 - $} \require{color}\colorbox{lime}{$ 132 $} \right|) \) \( 5 \cdot \left( 16 - 2^{4} \right) + 8 \cdot (4 - \left| \require{color}\colorbox{aqua}{$ 12 $} \right|) \) \( 5 \cdot \left( 16 - 2^{4} \right) + 8 \cdot (\require{color}\colorbox{violet}{$ 4 - $} \require{color}\colorbox{aqua}{$ 12 $}) \) \( 5 \cdot \left( 16 - 2^{4} \right) + \require{color}\colorbox{yellow}{$ 8 \cdot $} ( \require{color}\colorbox{violet}{$ - 8 $}) \) \( 5 \cdot \left( 16 - \require{color}\colorbox{lime}{$ 2^{4} $} \right) + ( \require{color}\colorbox{yellow}{$ - 64 $}) \) \( 5 \cdot \left( \require{color}\colorbox{aqua}{$ 16 - $} \require{color}\colorbox{lime}{$ 16 $} \right) + \left( - 64 \right) \) \( \require{color}\colorbox{pink}{$ 5 $} \cdot \require{color}\colorbox{aqua}{$ 0 $} + \left( - 64 \right) \) \( \require{color}\colorbox{pink}{$ 0 $} + \left( - 64 \right) \) \( - 64 \) |
Simplify \( - 14 + 3 \cdot \left( 18 - 10 \cdot \frac{3}{2} \right) - \left| {4.5}^{2} - 25 \right| \)
\( - 9.75 \)
If you need help arriving at this answer, click the solution button.
Begin working within the first grouping symbol. Follow the order of operations. |
\( - 14 + 3 \cdot \left( 18 - \require{color}\colorbox{yellow}{$ 10 \cdot \frac{3}{2} $} \right) - \left| {4.5}^{2} - 25 \right| \) \( - 14 + 3 \cdot \left( \require{color}\colorbox{lime}{$ 18 - $} \require{color}\colorbox{yellow}{$ 15 $} \right) - \left| {4.5}^{2} - 25 \right| \) \( - 14 + 3 \cdot \require{color}\colorbox{lime}{$ 3 $} - \left| \require{color}\colorbox{aqua}{$ {4.5}^{2} $} - 25 \right| \) \( - 14 + 3 \cdot 3 - \left| \require{color}\colorbox{aqua}{$ 20.25 $} \require{color}\colorbox{violet}{$ - 25 $} \right| \) \( - 14 + 3 \cdot 3 - \left| \require{color}\colorbox{violet}{$ - 4.75 $} \right| \) \( - 14 + \require{color}\colorbox{yellow}{$ 3 \cdot 3 $} - \require{color}\colorbox{violet}{$ 4.75 $} \) \( - 14 + \require{color}\colorbox{yellow}{$ 9 $} - 4.75 \) \( - 9.75 \) |
