## Simplifying Expressions using the Properties of Identities, Inverses, and Zero

Contents

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

## Example

Simplify: \(3x+15-3x.\)

### Solution

\(3x+15-3x\) | |

Notice the additive inverses, \(3x\) and \(-3x\). | \(0+15\) |

Add. | \(15\) |

## Example

Simplify: \(4\left(0.25q\right).\)

### Solution

\(4\left(0.25q\right)\) | |

Regroup, using the associative property. | \(\left[4\left(0.25\right)\right]q\) |

Multiply. | \(1.00q\) |

Simplify; 1 is the multiplicative identity. | \(q\) |

## Example

Simplify: \(\frac{0}{n+5}\), where \(n\ne -5\).

### Solution

\(\frac{0}{n+5}\) | |

Zero divided by any real number except itself is zero. | \(0\) |

## Example

Simplify: \(\frac{10-3p}{0}.\)

### Solution

\(\frac{10-3p}{0}\) | |

Division by zero is undefined. | undefined |

## Example

Simplify: \(\frac{3}{4}·\frac{4}{3}\left(6x+12\right).\)

### Solution

We cannot combine the terms in parentheses, so we multiply the two fractions first.

\(\frac{3}{4}·\frac{4}{3}\left(6x+12\right)\) | |

Multiply; the product of reciprocals is 1. | \(1\left(6x+12\right)\) |

Simplify by recognizing the multiplicative identity. | \(6x+12\) |

All the properties of real numbers we have used in this tutorial are summarized in the table below.

**Properties of Real Numbers**

Property | Of Addition | Of Multiplication |
---|---|---|

Commutative Property | ||

If a and b are real numbers then… | \(a+b=b+a\) | \(a·b=b·a\) |

Associative Property | ||

If a, b, and c are real numbers then… | \(\left(a+b\right)+c=a+\left(b+c\right)\) | \(\left(a·b\right)·c=a·\left(b·c\right)\) |

Identity Property | \(0\) is the additive identity | \(1\) is the multiplicative identity |

For any real number a, | \(\begin{array}{l}a+0=a\\ 0+a=a\end{array}\) | \(\begin{array}{l}a·1=a\\ 1·a=a\end{array}\) |

Inverse Property | \(-\mathit{\text{a}}\)is the additive inverse of \(a\) | \(a,a\ne 0\) \(1\mathit{\text{a}}\) is the |

For any real number a, | \(a+\text{(}\text{−}\mathit{\text{a}}\text{)}\phantom{\rule{0.2em}{0ex}}=0\) | \(a·1a=1\) |

Distributive Property\(\phantom{\rule{10em}{0ex}}\)If \(a,b,c\) are real numbers, then \(a\left(b+c\right)=ab+ac\) | ||

Properties of Zero | ||

For any real number a, | \(\begin{array}{l}a\cdot 0=0\\ 0\cdot a=0\end{array}\) | |

For any real number \(a,a\ne 0\) | \(\frac{0}{a}=0\) \(\)\(\frac{a}{0}\) is undefined |