Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as
x+5,43πr3x+5,\frac{4}{3}\pi {r}^{3}
, or
2m3n2\sqrt{2{m}^{3}{n}^{2}}
. In the expression
x+5x+5
, 5 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.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

(3)5=(3)(3)(3)(3)(3)x5=xxxxx\begin{matrix} \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\end{matrix}
(27)3=(27)(27)(27)(yz)3=(yz)(yz)(yz)\begin{matrix} \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{matrix}
In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

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.

Example 8: Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

  1. x + 5
  2. 43πr3\frac{4}{3}\pi {r}^{3}
  3. 2m3n2\sqrt{2{m}^{3}{n}^{2}}


Solution

Constants Variables
1. x + 5 5 x
2.
43πr3\frac{4}{3}\pi {r}^{3}
43,π\frac{4}{3},\pi
rr
3.
2m3n2\sqrt{2{m}^{3}{n}^{2}}
2
m,nm,n

Try It 8

List the constants and variables for each algebraic expression.

  1. 2πr(r+h)2\pi r\left(r+h\right)
  2. 2(L + W)
  3. 4y3+y4{y}^{3}+y


Solution

Example 9: Evaluating an Algebraic Expression at Different Values

Evaluate the expression
2x72x - 7
for each value for x.

  1. x=0x=0
  2. x=1x=1
  3. x=12x=\frac{1}{2}
  4. x=4x=-4


Solution

  1. Substitute 0 for
    xx
    .

    2x7=2(0)7=07=7\begin{matrix} 2x-7 & = 2\left(0\right)-7 \\ & =0-7 \\ & =-7\end{matrix}
  2. Substitute 1 for
    xx
    .

    2x7=2(1)7=27=5\begin{matrix} 2x-7 & = 2\left(1\right)-7 \\ & =2-7 \\ & =-5\end{matrix}
  3. Substitute
    12\frac{1}{2}
    for
    xx
    .

    2x7=2(12)7=17=6\begin{matrix} 2x-7 & = 2\left(\frac{1}{2}\right)-7 \\ & =1-7 \\ & =-6\end{matrix}
  4. Substitute
    4-4
    for
    xx
    .

    2x7=2(4)7=87=15\begin{matrix} 2x-7 & = 2\left(-4\right)-7 \\ & =-8-7 \\ & =-15\end{matrix}


Try It 9

Evaluate the expression
113y11 - 3y
for each value for y.

a.

y=2y=2


b.
y=0y=0


c.
y=23y=\frac{2}{3}


d.
y=5y=-5

Solution

Example 10: Evaluating Algebraic Expressions

Evaluate each expression for the given values.

  1. x+5x+5
    for
    x=5x=-5
  2. t2t1\frac{t}{2t - 1}\\
    for
    t=10t=10
  3. 43πr3\frac{4}{3}\pi {r}^{3}\\
    for
    r=5r=5
  4. a+ab+ba+ab+b
    for
    a=11,b=8a=11,b=-8
  5. 2m3n2\sqrt{2{m}^{3}{n}^{2}}
    for
    m=2,n=3m=2,n=3


Solution

  1. Substitute
    5-5
    for
    xx
    .

    x+5=(5)+5=0\begin{matrix} x+5&=\left(-5\right)+5 \\ &=0\end{matrix}
  2. Substitute 10 for
    tt
    .

    t2t1=(10)2(10)1=10201=1019\begin{matrix} \frac{t}{2t-1}& =\frac{\left(10\right)}{2\left(10\right)-1} \\ & =\frac{10}{20-1} \\ & =\frac{10}{19}\end{matrix}
  3. Substitute 5 for
    rr
    .

    43πr3=43π(5)3=43π(125)=5003π\begin{matrix} \frac{4}{3}\pi r^{3} & =\frac{4}{3}\pi\left(5\right)^{3} \\ & =\frac{4}{3}\pi\left(125\right) \\ & =\frac{500}{3}\pi\end{matrix}
  4. Substitute 11 for
    aa
    and –8 for
    bb
    .

    a+ab+b=(11)+(11)(8)+(8)=1188=85\begin{matrix} a+ab+b & =\left(11\right)+\left(11\right)\left(-8\right)+\left(-8\right) \\ & =11-8-8 \\ & =-85\end{matrix}
  5. Substitute 2 for
    mm
    and 3 for
    nn
    .

    2m3n2=2(2)3(3)2=2(8)(9)=144=12\begin{matrix} \sqrt{2m^{3}n^{2}} & =\sqrt{2\left(2\right)^{3}\left(3\right)^{2}} \\ & =\sqrt{2\left(8\right)\left(9\right)} \\ & =\sqrt{144} \\ & =12\end{matrix}


Try It 10

Evaluate each expression for the given values.

a.
y+3y3\frac{y+3}{y - 3}
for
y=5y=5


b.
72t7 - 2t
for
t=2t=-2


c.
13πr2\frac{1}{3}\pi {r}^{2}
for
r=11r=11


d.
(p2q)3{\left({p}^{2}q\right)}^{3}
for
p=2,q=3p=-2,q=3


e.
4(mn)5(nm)4\left(m-n\right)-5\left(n-m\right)
for
m=23,n=13m=\frac{2}{3},n=\frac{1}{3}


Solution

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