Simplifying-and-evaluating-expression

Mathematics

Simplifying Numerical Expressions and Evaluating Algebraic Expressions

Objectives

  • Simplify numerical expressions.

  • Evaluate algebraic expressions for given values of the variables.

Simplifying Numerical Expressions

  • To simplify a numerical expression means to replace the expression with a value resulting from performing the operations involved.

  • Order of Operations to follow when simplifying numerical expressions:

    1. Perform operations within grouping symbols (parentheses, brackets, braces, fraction line, radical sign).

      • For nested grouping symbols, simplify the innermost group first.

    2. Evaluate any exponential expressions.

    3. Multiply or divide from left to right.

    4. Add or subtract from left to right.

Example of Simplifying Numerical Expressions

  • Problem: Simplify: -2 + 3 [ (-1-1 + (-3 +2) )^3 ]^2

  • Solution Steps:

    • Start with the expression: -2 + 3[ (-1-1 + (-3 +2) )^3 ]^2

    • Simplify the grouping symbols:

      • -2 + 3 [ (-2 + (-1) )^3 ]^2

    • Evaluate powers:

      • -2 + 3 [ -8 + 1 ]

    • Add:

      • -2 + 3(-7)

    • Multiply:

      • -2 + (-21)

    • Final answer:

      • -23

Activity: Find My Value!

  • Assign numerical values to letters (A=1, B=2,..., Z=26).

  • Example: Calculate the value of a name:

    • Name: ANNE

      • A=1, N=14, N=14, E=5

      • Total: 1 + 14 + 14 + 5 = 34

  • Tasks:

    1. List five first names of close friends.

    2. Find the numerical value of each name.

Evaluating Algebraic Expressions

  • The process of replacing variables with certain numbers to find the value of the expression.

  • Uses the Substitution Property of Equality, which states: If two quantities are equal (a=b), one can be substituted for the other.

Important Notes

  • Replacing a variable in a polynomial gives it a numerical value. This process is called evaluating the polynomial.

Steps in Evaluating Algebraic Expressions

  1. Replace the variable with the given numerical value through substitution.

  2. Perform arithmetic according to order of operations:

    • Simplify expressions within grouping symbols.

    • Simplify powers.

    • Simplify products and quotients from left to right.

    • Simplify sums and differences from left to right.

Example Evaluations of Polynomials

  • Example 1: Evaluate when x=1 for 6x^2 + 3

    • 6(1)^2 + 3

    • Result: 9

  • Example 2: Evaluate when x=2 for 6x^2 + 3

    • 6(2)^2 + 3

    • Result: 27

  • Example 3: Evaluate when x=-3 for 6x^2 + 3

    • 6(-3)^2 + 3

    • Result: 57

Note on Multiple Variables

  • When an expression has more than one variable, a value for each variable is needed to find the numerical value of the polynomial.

Example with Two Variables

  • Problem: Evaluate when x=1, y=-1 for (4^−3)(3^−4)

    • Result: 1

  • Second Problem: Evaluate when x=2, y=3 for (4^−3)(3^−4)

    • Result: 5

Thank You for Listening!