Learning Outcomes
- Use the order of operations to simplify expressions that involve integer multiplication, division, addition, and subtraction
- Evaluate integer expressions using the order of operations
Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.
example
[latex]\text{Simplify: }7\left(-2\right)+4\left(-7\right)-6[/latex]
Solution:
We use the order of operations. Multiply first and then add and subtract from left to right.
|
[latex]7\left(-2\right)+4\left(-7\right)-6[/latex] |
| Multiply first. |
[latex]-14+\left(-28\right)-6[/latex] |
| Add. |
[latex]-42 - 6[/latex] |
| Subtract. |
[latex]-48[/latex] |
Watch the following video to see another example of how to use the order of operations to simplify an expression that contains integers.
In our next example we will simplify expressions with integers that also contain exponents.
example
Simplify:
- [latex]{\left(-2\right)}^{4}[/latex]
- [latex]{-2}^{4}[/latex]
Show Solution
Solution:
The exponent tells how many times to multiply the base.
1. The exponent is [latex]4[/latex] and the base is [latex]-2[/latex]. We raise [latex]-2[/latex] to the fourth power.
|
[latex]{\left(-2\right)}^{4}[/latex] |
| Write in expanded form. |
[latex]\left(-2\right)\left(-2\right)\left(-2\right)\left(-2\right)[/latex] |
| Multiply. |
[latex]4\left(-2\right)\left(-2\right)[/latex] |
| Multiply. |
[latex]-8\left(-2\right)[/latex] |
| Multiply. |
[latex]16[/latex] |
2. The exponent is [latex]4[/latex] and the base is [latex]2[/latex]. We raise [latex]2[/latex] to the fourth power and then take the opposite.
|
[latex]-{2}^{4}[/latex] |
| Write in expanded form. |
[latex]-\left(2\cdot 2\cdot 2\cdot 2\right)[/latex] |
| Multiply. |
[latex]-\left(4\cdot 2\cdot 2\right)[/latex] |
| Multiply. |
[latex]-\left(8\cdot 2\right)[/latex] |
| Multiply. |
[latex]-16[/latex] |
Now you try it.
example
[latex]\text{Simplify: }12 - 3\left(9 - 12\right)[/latex]
Show Solution
Solution:
According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.
|
[latex]12 - 3\left(9 - 12\right)[/latex] |
| Subtract the parentheses first. |
[latex]12 - 3\left(-3\right)[/latex] |
| Multiply. |
[latex]12-\left(-9\right)[/latex] |
| Subtract. |
[latex]\text{21}[/latex] |
example
Simplify: [latex]8\left(-9\right)\div {\left(-2\right)}^{3}[/latex]
Show Solution
Solution:
We simplify the exponent first, then multiply and divide.
|
[latex]8\left(-9\right)\div {\left(-2\right)}^{3}[/latex] |
| Simplify the exponent. |
[latex]8\left(-9\right)\div \left(-8\right)[/latex] |
| Multiply. |
[latex]-72\div \left(-8\right)[/latex] |
| Divide. |
[latex]\text{9}[/latex] |
example
[latex]\text{Simplify:}-30\div 2+\left(-3\right)\left(-7\right)[/latex]
Show Solution
Solution:
First we will multiply and divide from left to right. Then we will add.
|
[latex]-30\div 2+\left(-3\right)\left(-7\right)[/latex] |
| Divide. |
[latex]-15+\left(-3\right)\left(-7\right)[/latex] |
| Multiply. |
[latex]-15+21[/latex] |
| Add. |
[latex]\text{6}[/latex] |
In the following video we show more examples of how to evaluate expressions with integers using the order of operations.
Evaluate Variable Expressions with Integers
Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.
example
[latex]\text{Evaluate }2{x}^{2}-3x+8\text{ when }x=-4[/latex]
Show Solution
Solution:
|
[latex]2{x}^{2}-3x+8[/latex] |
| [latex]\text{Substitute}\color{red}{-4}\text{ for }x[/latex] |
[latex]2(\color{red}{-4})^{2}-3(\color{red}{-4})+8[/latex] |
| Simplify exponents. |
[latex]2(16)-3(-4)+8[/latex] |
| Multiply. |
[latex]32-(-12)+8[/latex] |
| Subtract. |
[latex]44+8[/latex] |
| Add. |
[latex]52[/latex] |
Keep in mind that when we substitute [latex]-4[/latex] for [latex]x[/latex], we use parentheses to show the multiplication. Without parentheses, it would look like [latex]2\cdot {-4}^{2}-3\cdot -4+8[/latex].
example
[latex]\text{Evaluate }3x+4y - 6\text{ when }x=-1\text{ and }y=2[/latex].
Show Solution
Solution:
|
[latex]3x+4y-6[/latex] |
| Substitute [latex]x=-1[/latex] and [latex]y=2[/latex] . |
[latex]3(\color{red}{-1})+4(\color{blue}{2})-6[/latex] |
| Multiply. |
[latex]-3+8-6[/latex] |
| Simplify. |
[latex]-1[/latex] |
In the following video we show more examples of how to substitute integers into variable expressions.
