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.	$15$

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example

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

Show Solution

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.	$q$

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example

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

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Solution:

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

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example

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

Show Solution

Solution:

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

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example

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

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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.	$6x+12$

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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

Module 8: Real Numbers