Simplifying and Evaluating Expressions with Integers That Require All Operations

Learning Outcomes

Simplify expressions using multiplication and division of integers
Evaluate variable expressions with multiplication and division of integers

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]

try it

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

Now you try it.

try it

example

[latex]\text{Simplify: }12 - 3\left(9 - 12\right)[/latex]

Show Solution

try it

example

Simplify: [latex]8\left(-9\right)\div {\left(-2\right)}^{3}[/latex]

Show Solution

try it

example

[latex]\text{Simplify:}-30\div 2+\left(-3\right)\left(-7\right)[/latex]

Show Solution

try it

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

try it

example

[latex]\text{Evaluate }3x+4y - 6\text{ when }x=-1\text{ and }y=2[/latex].

Show Solution

try it

In the following video we show more examples of how to substitute integers into variable expressions.

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	[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]

	[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]

	[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]

	[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]

	[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]

	[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]

	[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]

Module 3: Integers