Evaluate Algebraic Expressions
Learning Outcomes
- Define and identify constants in an algebraic expression
- Evaluate algebraic expressions for different values
[latex]\begin{array}{ll}\left(-3\right)^{5}=\left(-3\right)\cdot\left(-3\right)\cdot\left(-3\right)\cdot\left(-3\right)\cdot\left(-3\right)\,\,\, & x^{5}=x\cdot x\cdot x\cdot x\cdot x \\ \left(2\cdot7\right)^{3}=\left(2\cdot7\right)\cdot\left(2\cdot7\right)\cdot\left(2\cdot7\right) & \left(yz\right)^{3}=\left(yz\right)\cdot\left(yz\right)\cdot\left(yz\right)\end{array}[/latex]
In each case, the exponent tells us how many factors of the base to use regardless of whether the base consists of constants or variables.
In the following example, we will practice identifying constants and variables in mathematical expressions.
Example
List the constants and variables for each algebraic expression.- [latex]x+5[/latex]
- [latex]\dfrac{4}{3}\normalsize\pi {r}^{3}[/latex]
- [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex]
Answer:
Expression | Constants | Variables |
---|---|---|
1. [latex] x + 5 [/latex] | [latex]5[/latex] | [latex]x[/latex] |
2. [latex]\dfrac{4}{3}\normalsize\pi {r}^{3}[/latex] | [latex]\dfrac{4}{3}\normalsize,\pi [/latex] | [latex]r[/latex] |
3. [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex] | [latex]2[/latex] | [latex]m,n[/latex] |
Example
Evaluate the expression [latex]2x - 7[/latex] for each value for [latex]x[/latex].- [latex]x=0[/latex]
- [latex]x=1[/latex]
- [latex]x=\dfrac{1}{2}[/latex]
- [latex]x=-4[/latex]
Answer:
- Substitute 0 for [latex]x[/latex].
[latex]\begin{array}{ll}2x-7 \hfill& = 2\left(0\right)-7 \\ \hfill& =0-7 \\ \hfill& =-7\end{array}[/latex]
- Substitute 1 for [latex]x[/latex].
[latex]\begin{array}{ll}2x-7 \hfill& = 2\left(1\right)-7 \\ \hfill& =2-7 \\ \hfill& =-5\end{array}[/latex]
- Substitute [latex]\frac{1}{2}[/latex] for [latex]x[/latex].
[latex]\begin{array}{ll}2x-7 \hfill& = 2\left(\frac{1}{2}\right)-7 \\ \hfill& =1-7 \\ \hfill& =-6\end{array}[/latex]
- Substitute [latex]-4[/latex] for [latex]x[/latex].
[latex]\begin{array}{ll}2x-7 \hfill& = 2\left(-4\right)-7 \\ \hfill& =-8-7 \\ \hfill& =-15\end{array}[/latex]
Example
Evaluate each expression for the given values.- [latex]x+5[/latex] for [latex]x=-5[/latex]
- [latex]\dfrac{t}{2t - 1}[/latex] for [latex]t=10[/latex]
- [latex]\dfrac{4}{3}\normalsize\pi {r}^{3}[/latex] for [latex]r=5[/latex]
- [latex]a+ab+b[/latex] for [latex]a=11,b=-8[/latex]
- [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex] for [latex]m=2,n=3[/latex]
Answer:
- Substitute [latex]-5[/latex] for [latex]x[/latex].
[latex]\begin{array}{ll}x+5\hfill&=\left(-5\right)+5 \\ \hfill&=0\end{array}[/latex]
- Substitute 10 for [latex]t[/latex].
[latex]\begin{array}{ll}\frac{t}{2t-1}\hfill& =\frac{\left(10\right)}{2\left(10\right)-1} \\ \hfill& =\frac{10}{20-1} \\ \hfill& =\frac{10}{19}\end{array}[/latex]
- Substitute 5 for [latex]r[/latex].
[latex]\begin{array}{ll}\frac{4}{3}\pi r^{3} \hfill& =\frac{4}{3}\pi\left(5\right)^{3} \\ \hfill& =\frac{4}{3}\pi\left(125\right) \\ \hfill& =\frac{500}{3}\pi\end{array}[/latex]
- Substitute 11 for [latex]a[/latex] and –8 for [latex]b[/latex].
[latex]\begin{array}{ll}a+ab+b \hfill& =\left(11\right)+\left(11\right)\left(-8\right)+\left(-8\right) \\ \hfill& =11-88-8 \\ \hfill& =-85\end{array}[/latex]
- Substitute 2 for [latex]m[/latex] and 3 for [latex]n[/latex].
[latex]\begin{array}{ll}\sqrt{2m^{3}n^{2}} \hfill& =\sqrt{2\left(2\right)^{3}\left(3\right)^{2}} \\ \hfill& =\sqrt{2\left(8\right)\left(9\right)} \\ \hfill& =\sqrt{144} \\ \hfill& =12\end{array}[/latex]
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Evaluate Various Algebraic Expressions. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Specific attribution
- College Algebra: Evaluating Algebraic Expressions. License: CC BY: Attribution.