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Study Guides > Intermediate Algebra

Evaluate Algebraic Expressions

Learning Outcomes

  • Define and identify constants in an algebraic expression
  • Evaluate algebraic expressions for different values
In mathematics, we may see expressions such as [latex]x+5,\dfrac{4}{3}\normalsize\pi {r}^{3}[/latex], or [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex]. In the expression [latex]x+5[/latex], [latex]5[/latex] is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. The examples in this section include exponents. Recall that an exponent is shorthand for writing repeated multiplication of the same number. When variables have exponents, it means repeated multiplication of the same variable. The base of an exponent is the number or variable being multiplied, and the exponent tells us how many times to multiply. For example,
[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.
  1. [latex]x+5[/latex]
  2. [latex]\dfrac{4}{3}\normalsize\pi {r}^{3}[/latex]
  3. [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]

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before. In the next example we show how to substitute various types of numbers into a mathematical expression.

Example

Evaluate the expression [latex]2x - 7[/latex] for each value for [latex]x[/latex].
  1. [latex]x=0[/latex]
  2. [latex]x=1[/latex]
  3. [latex]x=\dfrac{1}{2}[/latex]
  4. [latex]x=-4[/latex]

Answer:

  1. 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]
  2. 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]
  3. 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]
  4. 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]

Now we will show more examples of evaluating a variety of mathematical expressions for various values.

Example

Evaluate each expression for the given values.
  1. [latex]x+5[/latex] for [latex]x=-5[/latex]
  2. [latex]\dfrac{t}{2t - 1}[/latex] for [latex]t=10[/latex]
  3. [latex]\dfrac{4}{3}\normalsize\pi {r}^{3}[/latex] for [latex]r=5[/latex]
  4. [latex]a+ab+b[/latex] for [latex]a=11,b=-8[/latex]
  5. [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex] for [latex]m=2,n=3[/latex]

Answer:

  1. Substitute [latex]-5[/latex] for [latex]x[/latex].
    [latex]\begin{array}{ll}x+5\hfill&=\left(-5\right)+5 \\ \hfill&=0\end{array}[/latex]
  2. 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]
  3. 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]
  4. 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]
  5. 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]

In the following video we present more examples of evaluating a variety of expressions for given values. https://youtu.be/MkRdwV4n91g

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  • 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.

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