Algebraic Operations on Functions: Background You’ll Need 1

  • Simplify and calculate an algebraic equation

Algebraic Expressions

An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. For example, [latex]3x + 2y - 7[/latex] is an algebraic expression that contains two variables [latex]x[/latex] and [latex]y[/latex] and three constants [latex]3[/latex], [latex]2[/latex], and [latex]7[/latex].

constant, variable, algebraic expression

  • A constant is a fixed value or a number that does not change in a particular context.
  • A variable is a symbol that represents a value or quantity that can change or vary in a given situation or context.
  • An algebraic expression is a mathematical phrase or combination of numbers, variables, and arithmetic operations such as 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.

[latex]\begin{align}&\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)\\ \text{ }\end{align}[/latex]

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables. When naming the variable, ignore any exponents or radicals containing the variable.

List the constants and variables for each algebraic expression.

  1. [latex]x + 5[/latex]
  2. [latex]\frac{4}{3}\pi {r}^{3}[/latex]
  3. [latex]\sqrt{2{m}^{3}{n}^{2}}[/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.

How To: Evaluate Algebraic Expressions
[latex]\\[/latex]
Use the following steps to evaluate an algebraic expression:

  1. Replace each variable in the expression with the given value
  2. Simplify the resulting expression using the order of operations

Note: If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Be Careful when simplifying fractions! Why does the fraction [latex]\dfrac{(25)}{3(25)-1}[/latex] not simplify to [latex]\dfrac{\cancel{(25)}}{3\cancel{(25)}-1}=\dfrac{1}{3-1}=\dfrac{1}{2}[/latex]?

Using the inverse property of multiplication, we are permitted to “cancel out” common factors in the numerator and denominator such that [latex]\dfrac{a}{a}=1[/latex].But be careful! We have no rule that allows us to cancel numbers in the top and bottom of a fractions that are contained in sums or differences. You’ll see this idea reappear frequently throughout the course.

Evaluate each expression for the given values.

  1. [latex]x+5[/latex] for [latex]x=-5[/latex]
  2. [latex]\frac{t}{2t - 1}[/latex] for [latex]t=10[/latex]
  3. [latex]\dfrac{4}{3}\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]