## Simplifying Expressions with Integers

Contents

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:

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

### 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. | \(\text{6}\) |