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: [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] |
try it
[ohm_question]146488[/ohm_question]example
Simplify: [latex]4\left(0.25q\right)[/latex]Answer: Solution:
| [latex]4\left(0.25q\right)[/latex] | |
| Regroup, using the associative property. | [latex]\left[4\left(0.25\right)\right]q[/latex] |
| Multiply. | [latex]1.00q[/latex] |
| Simplify; 1 is the multiplicative identity. | [latex]q[/latex] |
try it
[ohm_question]146489[/ohm_question]example
Simplify: [latex]{\Large\frac{0}{n+5}}[/latex] , where [latex]n\ne -5[/latex]Answer: Solution:
| [latex]{\Large\frac{0}{n+5}}[/latex] | |
| Zero divided by any real number except itself is zero. | [latex]0[/latex] |
try it
[ohm_question]146490[/ohm_question]example
Simplify: [latex]{\Large\frac{10 - 3p}{0}}[/latex].Answer: Solution:
| [latex]{\Large\frac{10 - 3p}{0}}[/latex] | |
| Division by zero is undefined. | undefined |
try it
[ohm_question]146491[/ohm_question]example
Simplify: [latex]{\Large\frac{3}{4}}\cdot {\Large\frac{4}{3}}\left(6x+12\right)[/latex].Answer: Solution: We cannot combine the terms in parentheses, so we multiply the two fractions first.
| [latex]{\Large\frac{3}{4}}\cdot {\Large\frac{4}{3}}\left(6x+12\right)[/latex] | |
| Multiply; the product of reciprocals is 1. | [latex]1\left(6x+12\right)[/latex] |
| Simplify by recognizing the multiplicative identity. | [latex]6x+12[/latex] |
try it
[ohm_question]146493[/ohm_question]| 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 | |
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- Question ID 146493, 146491, 146490, 146487. Authored by: Lumen Learning. License: CC BY: Attribution.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
