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: [latex]3x+15 - 3x[/latex]
Solution:
| [latex]3x+15 - 3x[/latex] | |
| Notice the additive inverses, [latex]3x[/latex] and [latex]-3x[/latex] . | [latex]0+15[/latex] |
| Add. | [latex]15[/latex] |
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example
Simplify: [latex]4\left(0.25q\right)[/latex]
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example
Simplify: [latex]{\Large\frac{0}{n+5}}[/latex] , where [latex]n\ne -5[/latex]
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example
Simplify: [latex]{\Large\frac{10 - 3p}{0}}[/latex].
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example
Simplify: [latex]{\Large\frac{3}{4}}\cdot {\Large\frac{4}{3}}\left(6x+12\right)[/latex].
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All the properties of real numbers we have used in this chapter are summarized in the table below.
| Property | Of Addition | Of Multiplication |
|---|---|---|
| Commutative Property | ||
| If a and b are real numbers then… | [latex]a+b=b+a[/latex] | [latex]a\cdot b=b\cdot a[/latex] |
| Associative Property | ||
| If a, b, and c are real numbers then… | [latex]\left(a+b\right)+c=a+\left(b+c\right)[/latex] | [latex]\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)[/latex] |
| Identity Property | [latex]0[/latex] is the additive identity | [latex]1[/latex] is the multiplicative identity |
| For any real number a, | [latex]\begin{array}{l}a+0=a\\ 0+a=a\end{array}[/latex] | [latex]\begin{array}{l}a\cdot 1=a\\ 1\cdot a=a\end{array}[/latex] |
| Inverse Property | [latex]-\mathit{\text{a}}[/latex] is the additive inverse of [latex]a[/latex] | [latex]a,a\ne 0[/latex]
[latex]\frac{1}{a}[/latex] is the multiplicative inverse of [latex]a[/latex] |
| For any real number a, | [latex]a+\text{(}\text{-}\mathit{\text{a}}\text{)}=0[/latex] | [latex]a\cdot 1a=1[/latex] |
| Distributive Property
If [latex]a,b,c[/latex] are real numbers, then [latex]a\left(b+c\right)=ab+ac[/latex] |
||
| Properties of Zero | ||
| For any real number a, | [latex]\begin{array}{l}a\cdot 0=0\\ 0\cdot a=0\end{array}[/latex] | |
| For any real number [latex]a,a\ne 0[/latex] | [latex]{\Large\frac{0}{a}}=0[/latex]
[latex]{\Large\frac{a}{0}}[/latex] is undefined |
|
