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Study Guides > Prealgebra

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+15−3x3x+15 - 3x. Solution:
3x+15−3x3x+15 - 3x
Notice the additive inverses, 3x3x and −3x-3x . 0+150+15
Add. 1515

try it

[ohm_question]146488[/ohm_question]

example

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

Answer: Solution:

4(0.25q)4\left(0.25q\right)
Regroup, using the associative property. [4(0.25)]q\left[4\left(0.25\right)\right]q
Multiply. 1.00q1.00q
Simplify; 1 is the multiplicative identity. qq

try it

[ohm_question]146489[/ohm_question]

example

Simplify: 0n+5\frac{0}{n+5} , where n≠−5n\ne -5.

Answer: Solution:

0n+5\frac{0}{n+5}
Zero divided by any real number except itself is zero. 00

try it

[ohm_question]146490[/ohm_question]

example

Simplify: 10−3p0\frac{10 - 3p}{0}.

Answer: Solution:

10−3p0\frac{10 - 3p}{0}
Division by zero is undefined. undefined

try it

[ohm_question]146491[/ohm_question]

example

Simplify: 34⋅43(6x+12)\frac{3}{4}\cdot \frac{4}{3}\left(6x+12\right).

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

34⋅43(6x+12)\frac{3}{4}\cdot \frac{4}{3}\left(6x+12\right)
Multiply; the product of reciprocals is 1. 1(6x+12)1\left(6x+12\right)
Simplify by recognizing the multiplicative identity. 6x+126x+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+aa+b=b+a a⋅b=b⋅aa\cdot b=b\cdot a
Associative Property
If a, b, and c are real numbers then… (a+b)+c=a+(b+c)\left(a+b\right)+c=a+\left(b+c\right) (a⋅b)⋅c=a⋅(b⋅c)\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)
Identity Property 00 is the additive identity 11 is the multiplicative identity
For any real number a, a+0=a0+a=a\begin{array}{l}a+0=a\\ 0+a=a\end{array} a⋅1=a1⋅a=a\begin{array}{l}a\cdot 1=a\\ 1\cdot a=a\end{array}
Inverse Property −a-\mathit{\text{a}} is the additive inverse of aa a,a≠0a,a\ne 0 1a\frac{1}{a} is the multiplicative inverse of aa
For any real number a, a+(-a)=0a+\text{(}\text{-}\mathit{\text{a}}\text{)}=0 a⋅1a=1a\cdot 1a=1
Distributive Property If a,b,ca,b,c are real numbers, then a(b+c)=ab+aca\left(b+c\right)=ab+ac
Properties of Zero
For any real number a, a⋅0=00⋅a=0\begin{array}{l}a\cdot 0=0\\ 0\cdot a=0\end{array}
For any real number a,a≠0a,a\ne 0 0a=0\frac{0}{a}=0 a0\frac{a}{0} is undefined

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