Mathematics » Properties of Real Numbers » Properties of Identity, Inverses, and Zero

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

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

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

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 multiplicative inverse of $$a$$

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

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