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Simplifying Expressions with Integers

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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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