## Evaluating Expressions using the Commutative and Associative Properties

The commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier, as the next several examples illustrate.

## Example

Evaluate each expression when \(x=\frac{7}{8}.\)

- \(\phantom{\rule{0.2em}{0ex}}x+0.37+\left(-x\right)\)
- \(\phantom{\rule{0.2em}{0ex}}x+\left(-x\right)+0.37\)

### Solution

(1) | |

Substitute \(\frac{7}{8}\) for \(x\). | |

Convert fractions to decimals. | |

Add left to right. | |

Subtract. |

(2) | |

Substitute \(\frac{7}{8}\) for x. | |

Add opposites first. |

What was the difference between part and part ? Only the order changed. By the Commutative Property of Addition, \(x+0.37+\left(-x\right)=x+\left(-x\right)+0.37.\) But wasn’t part much easier?

Let’s do one more, this time with multiplication.

## Example

Evaluate each expression when \(n=17.\)

- \(\phantom{\rule{0.2em}{0ex}}\frac{4}{3}\left(\frac{3}{4}n\right)\)
- \(\phantom{\rule{0.2em}{0ex}}\left(\frac{4}{3}·\frac{3}{4}\right)n\)

### Solution

(1) | |

Substitute 17 for n. | |

Multiply in the parentheses first. | |

Multiply again. |

(2) | |

Substitute 17 for n. | |

Multiply. The product of reciprocals is 1. | |

Multiply again. |

What was the difference between part and part here? Only the grouping changed. By the Associative Property of Multiplication, \(\frac{4}{3}\left(\frac{3}{4}n\right)=\left(\frac{4}{3}·\frac{3}{4}\right)n.\) By carefully choosing how to group the factors, we can make the work easier.