Evaluating Algebraic Expressions Notes

Evaluating Algebraic Expressions

Objectives

  • Recall the rules in operations and integers.
  • PEMDAS
  • Exponential notation
  • Evaluate mathematical expressions for given variable values.

Exponential Notation

  • If 9 cubed = 729
    • Base: the number being multiplied.
    • Exponent: tells how many times the base is used as a factor.
    • Power: the product of equal factors.
  • Example: 9^3 = 9 * 9 * 9 = 729

Reading Exponential Notation

  • x^4: x raised to the fourth power or x to the fourth power
  • y^2: y squared, y raised to two, or y to the second power

Expanding Notation

  • x^4 means x * x * x * x
  • y^2 means y * y
  • Example: 6 * 6 = 36
  • Fraction Example:
    • (-2/3)^4 = (-2/3) * (-2/3) * (-2/3) * (-2/3) = 16/81

Parentheses and Negatives

  • (-2)^4 = 16 (2 * 2 * 2 * 2 = 16, but with negatives, it becomes positive)
  • Example:
    • (2/3)^4 = (2 * 2 * 2 * 2) / (3 * 3 * 3 * 3) = 16/81
  • Raising it to the fifth power:
    • (-1/2)^5 = (-1 * -1 * -1 * -1 * -1) / (2 * 2 * 2 * 2 * 2) = -1/32

More Expanding

  • 3^3 / 4^2 = (3 * 3 * 3) / (4 * 4) = 27/16 = 432

Evaluating Algebraic Expressions

  • Replacing a variable with a particular value and simplifying the expression.
  • Substitution: Substituting the value for the variable and finding the value of the expression.

Examples

  • If 3x and x = 3, then 3x = 3 * 3 = 9
  • Two variables: If 3xy, x = 3, and y = -2, then 3xy = 3 * 3 * -2 = -18
  • With Addition: If 3x + y, x = 4, and y = 5, then 3x + y = (3 * 4) + 5 = 12 + 5 = 17

More Examples with Given Values

  • If a = 2, b = 3, and c = -2

Example 1

  • 5a = 5 * 2 = 10

Example 2

  • ab = a * b = 2 * 3 = 6

Example 3

  • abc = 2 * 3 * -2 = 6 * -2 = -12

Example 4

  • -5b - c = (-5 * 3) - (-2) = -15 + 2 = -13

Example 5

  • a(b + c) = 2(3 + (-2)) = 2(1) = 2

Example 6

  • b(4a - 5c) = 3(4(2) - 5(-2)) = 3(8 + 10) = 3(18) = 54

Example 7

  • 8b / ac = (8 * 3) / (2 * -2) = 24 / -4 = -6

Examples with Exponents

  • If x = 3, y = -2, and z = 4

Example 1

  • xy^3 = 3 * (-2)^3 = 3 * (-2 * -2 * -2) = 3 * -8 = -24

Example 2

  • 2x^2 + 4y - z = 2(3)^2 + 4(-2) - 4 = 2(9) - 8 - 4 = 18 - 8 - 4 = 6

Example 3

  • (3x^3 + 1) / (z - 2) = (3(3^3) + 1) / (4 - 2) = (3(27) + 1) / 2 = (81 + 1) / 2 = 82 / 2 = 41

Example 4

  • (2/3)x^2 - 3y = (2/3)(3^2) - 3(-2) = (2/3)(9) + 6 = 6 + 6 = 12

Examples with Fractional Values

  • If x = 2/3 and y = -1/2

Example 1

  • 3xy^3 = 3 * (2/3) * (-1/2)^3 = 3 * (2/3) * (-1/8) = -1/4

Example 2

  • (3x^3 + 4) / z^2, z = -1/2 = (3 * (2/3)^3 + 4) / (-1/2)^2 = (3 * (8/27) + 4) / (1/4)
  • (8/9 + 4) / (1/4) = (44/9) / (1/4) = (44/9) * (4/1) = 176/9 = -39 remainder 1