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

### Learning Outcomes

• Simplify algebraic expressions using identity, inverse and zero properties
• Identify which property(ies) to use to simplify an algebraic expression

## Simplify 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. $15$

### try it

[ohm_question]146488[/ohm_question]

### example

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

Answer: 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. $q$

### try it

[ohm_question]146489[/ohm_question]

### example

Simplify: ${\Large\frac{0}{n+5}}$ , where $n\ne -5$

Answer: Solution:

 ${\Large\frac{0}{n+5}}$ Zero divided by any real number except itself is zero. $0$

### try it

[ohm_question]146490[/ohm_question]

### example

Simplify: ${\Large\frac{10 - 3p}{0}}$.

Answer: Solution:

 ${\Large\frac{10 - 3p}{0}}$ Division by zero is undefined. undefined

### try it

[ohm_question]146491[/ohm_question]

### example

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

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

 ${\Large\frac{3}{4}}\cdot {\Large\frac{4}{3}}\left(6x+12\right)$ Multiply; the product of reciprocals is 1. $1\left(6x+12\right)$ Simplify by recognizing the multiplicative identity. $6x+12$

### try it

[ohm_question]146493[/ohm_question]
All the properties of real numbers we have used in this chapter 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… $a+b=b+a$ $a\cdot b=b\cdot a$
Associative Property
If a, b, and c are real numbers then… $\left(a+b\right)+c=a+\left(b+c\right)$ $\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$
Identity Property $0$ is the additive identity $1$ is the multiplicative identity
For any real number a, $\begin{array}{l}a+0=a\\ 0+a=a\end{array}$ $\begin{array}{l}a\cdot 1=a\\ 1\cdot a=a\end{array}$
Inverse Property $-\mathit{\text{a}}$ is the additive inverse of $a$ $a,a\ne 0$ $\frac{1}{a}$ is the multiplicative inverse of $a$
For any real number a, $a+\text{(}\text{-}\mathit{\text{a}}\text{)}=0$ $a\cdot 1a=1$
Distributive Property 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 $a,a\ne 0$ ${\Large\frac{0}{a}}=0$ ${\Large\frac{a}{0}}$ is undefined

## Licenses & Attributions

### CC licensed content, Original

• Question ID 146493, 146491, 146490, 146487. Authored by: Lumen Learning. License: CC BY: Attribution.