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: 93=9∗9∗9=7299^3 = 9 * 9 * 9 = 729

Reading Exponential Notation

  • x4x^4: x raised to the fourth power or x to the fourth power
  • y2y^2: y squared, y raised to two, or y to the second power

Expanding Notation

  • x4x^4 means x * x * x * x
  • y2y^2 means y * y
  • Example: 6 * 6 = 36
  • Fraction Example:
    • (−2/3)4=(−2/3)∗(−2/3)∗(−2/3)∗(−2/3)=16/81(-2/3)^4 = (-2/3) * (-2/3) * (-2/3) * (-2/3) = 16/81

Parentheses and Negatives

  • (−2)4=16(-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(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(-1/2)^5 = (-1 * -1 * -1 * -1 * -1) / (2 * 2 * 2 * 2 * 2) = -1/32

More Expanding

  • 33/42=(3∗3∗3)/(4∗4)=27/16=4323^3 / 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 3x3x and x=3x = 3, then 3x=3∗3=93x = 3 * 3 = 9
  • Two variables: If 3xy3xy, x=3x = 3, and y=−2y = -2, then 3xy=3∗3∗−2=−183xy = 3 * 3 * -2 = -18
  • With Addition: If 3x+y3x + y, x=4x = 4, and y=5y = 5, then 3x+y=(3∗4)+5=12+5=173x + y = (3 * 4) + 5 = 12 + 5 = 17

More Examples with Given Values

  • If a=2a = 2, b=3b = 3, and c=−2c = -2
Example 1
  • 5a=5∗2=105a = 5 * 2 = 10
Example 2
  • ab=a∗b=2∗3=6ab = a * b = 2 * 3 = 6
Example 3
  • abc=2∗3∗−2=6∗−2=−12abc = 2 * 3 * -2 = 6 * -2 = -12
Example 4
  • −5b−c=(−5∗3)−(−2)=−15+2=−13-5b - c = (-5 * 3) - (-2) = -15 + 2 = -13
Example 5
  • a(b+c)=2(3+(−2))=2(1)=2a(b + c) = 2(3 + (-2)) = 2(1) = 2
Example 6
  • b(4a−5c)=3(4(2)−5(−2))=3(8+10)=3(18)=54b(4a - 5c) = 3(4(2) - 5(-2)) = 3(8 + 10) = 3(18) = 54
Example 7
  • 8b/ac=(8∗3)/(2∗−2)=24/−4=−68b / ac = (8 * 3) / (2 * -2) = 24 / -4 = -6

Examples with Exponents

  • If x=3x = 3, y=−2y = -2, and z=4z = 4
Example 1
  • xy3=3∗(−2)3=3∗(−2∗−2∗−2)=3∗−8=−24xy^3 = 3 * (-2)^3 = 3 * (-2 * -2 * -2) = 3 * -8 = -24
Example 2
  • 2x2+4y−z=2(3)2+4(−2)−4=2(9)−8−4=18−8−4=62x^2 + 4y - z = 2(3)^2 + 4(-2) - 4 = 2(9) - 8 - 4 = 18 - 8 - 4 = 6
Example 3
  • (3x3+1)/(z−2)=(3(33)+1)/(4−2)=(3(27)+1)/2=(81+1)/2=82/2=41(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)x2−3y=(2/3)(32)−3(−2)=(2/3)(9)+6=6+6=12(2/3)x^2 - 3y = (2/3)(3^2) - 3(-2) = (2/3)(9) + 6 = 6 + 6 = 12

Examples with Fractional Values

  • If x=2/3x = 2/3 and y=−1/2y = -1/2
Example 1
  • 3xy3=3∗(2/3)∗(−1/2)3=3∗(2/3)∗(−1/8)=−1/43xy^3 = 3 * (2/3) * (-1/2)^3 = 3 * (2/3) * (-1/8) = -1/4
Example 2
  • (3x3+4)/z2,z=−1/2=(3∗(2/3)3+4)/(−1/2)2=(3∗(8/27)+4)/(1/4)(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(8/9 + 4) / (1/4) = (44/9) / (1/4) = (44/9) * (4/1) = 176/9 = -39 remainder 1