Mathematics » Introducing Integers » Multiply and Divide Integers

Simplifying Expressions with Integers

Simplifying Expressions with 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

\(\text{Simplify:}\phantom{\rule{0.2em}{0ex}}7\left(-2\right)+4\left(-7\right)-6.\)

Solution

We use the order of operations. Multiply first and then add and subtract from left to right.

 \(7\left(-2\right)+4\left(-7\right)-6\)
Multiply first.\(-14+\left(-28\right)-6\)
Add.\(-42-6\)
Subtract.\(-48\)

Example

Simplify:

  1. \((−2)^4\)
  2. \(\phantom{\rule{0.2em}{0ex}}{-2}^{4}\)

Solution

The exponent tells how many times to multiply the base.

The exponent is \(4\) and the base is \(-2.\) We raise \(-2\) to the fourth power.

 \({\left(-2\right)}^{4}\)
Write in expanded form.\(\left(-2\right)\left(-2\right)\left(-2\right)\left(-2\right)\)
Multiply.\(4\left(-2\right)\left(-2\right)\)
Multiply.\(-8\left(-2\right)\)
Multiply.\(16\)

The exponent is \(4\) and the base is \(2.\) We raise \(2\) to the fourth power and then take the opposite.

 \(-{2}^{4}\)
Write in expanded form.\(-\left(2\cdot 2\cdot 2\cdot 2\right)\)
Multiply.\(-\left(4\cdot 2\cdot 2\right)\)
Multiply.\(-\left(8\cdot 2\right)\)
Multiply.\(-16\)

Example

\(\text{Simplify:}\phantom{\rule{0.2em}{0ex}}12-3\left(9-12\right).\)

Solution

According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.

 \(12-3\left(9-12\right)\)
Subtract the parentheses first.\(12-3\left(-3\right)\)
Multiply.\(12-\left(-9\right)\)
Subtract.\(\text{21}\)

Example

Simplify: \(8\left(-9\right)÷{\left(-2\right)}^{3}.\)

Solution

We simplify the exponent first, then multiply and divide.

 \(8\left(-9\right)÷{\left(-2\right)}^{3}\)
Simplify the exponent.\(8\left(-9\right)÷\left(-8\right)\)
Multiply.\(-72÷\left(-8\right)\)
Divide.\(\text{9}\)

Example

\(\text{Simplify:}\phantom{\rule{0.2em}{0ex}}-30÷2+\left(-3\right)\left(-7\right).\)

Solution

First we will multiply and divide from left to right. Then we will add.

 \(-30÷2+\left(-3\right)\left(-7\right)\)
Divide.\(-15+\left(-3\right)\left(-7\right)\)
Multiply.\(-15+21\)
Add.\(\text{6}\)

Optional Video: Multiplying and Dividing Signed Numbers

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