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Mathematics » Introducing Integers » Multiply and Divide Integers

# 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. $$\text{6}$$