Sequences and Series: Background You’ll Need 1

  • Understand and simplify math expressions

Algebraic Expressions

In mathematics, we may see expressions such as [latex]x+5,\frac{4}{3}\pi {r}^{3}[/latex], or [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex]. In the expression [latex]x+5, 5[/latex] is called a constant because it does not vary and x is called a variable because it does. 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 ExpressionsUse 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!
[latex]\\[/latex]
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]?
[latex]\\[/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].
[latex]\\[/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]

Simplify Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

When simplifying algebraic expressions, we may sometimes need to add, subtract, simplify, multiply, or divide fractions. It is important to be able to do these operations on the fractions without converting them to decimals.
[latex]\\[/latex]
To multiply fractions, multiply the numerators and place them over the product of the denominators.

[latex]\dfrac{a}{b}\cdot\dfrac{c}{d} = \dfrac {ac}{bd}[/latex]

To divide fractions, multiply the first by the reciprocal of the second.

 [latex]\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}[/latex]

To simplify fractions, find common factors in the numerator and denominator that cancel.

 [latex]\dfrac{24}{32}=\dfrac{2\cdot2\cdot2\cdot3}{2\cdot2\cdot2\cdot2\cdot2}=\dfrac{3}{2\cdot2}=\dfrac{3}{4}[/latex]

To add or subtract fractions, first rewrite each fraction as an equivalent fraction such that each has a common denominator, then add or subtract the numerators and place the result over the common denominator.

 [latex]\dfrac{a}{b}\pm\dfrac{c}{d} = \dfrac{ad \pm bc}{bd}[/latex]

Simplify the following algebraic expressions:

  1. [latex]3x - 2y+x - 3y - 7[/latex]
  2. [latex]2r - 5\left(3-r\right)+4[/latex]
  3. [latex]\left(4t-\dfrac{5}{4}s\right)-\left(\dfrac{2}{3}t+2s\right)[/latex]
  4. [latex]2mn - 5m+3mn+n[/latex]