Algebraic Expressions and Evaluation

Evaluating Algebraic Expressions

Algebraic expressions can be evaluated by substituting given values for variables and performing the indicated operations.

• Key Point: Substitute the value of the variable into the expression and simplify.

• Example: Evaluate for .

  • Substitute:

  • Simplify:

Exponents and Their Properties

Laws of Exponents

The laws of exponents allow us to simplify expressions involving powers. When simplifying, always express answers with positive exponents.

• Product of Powers:

• Quotient of Powers:

• Power of a Power:

• Negative Exponent:

• Example: Simplify using positive exponents only.

  • First, apply the power rule:

  • Combine:

Polynomial Operations

Multiplying Binomials

Multiplying binomials can be done using the distributive property or special formulas such as the FOIL method.

• General Formula:

• Example:

  • Apply the difference of squares:

  • Here, ,

  • Result:

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four terms by grouping pairs of terms and factoring out common factors.

• Steps:

  1. Group terms in pairs.

  2. Factor out the greatest common factor (GCF) from each pair.

  3. If a common binomial factor appears, factor it out.

• Example: Factor by grouping.

  • Group:

  • Factor:

  • Factor out :

Radicals and Rational Expressions

Multiplying Radical Expressions

To multiply radical expressions, use the property and simplify if possible.

• Example: Multiply

  • Combine under one radical:

Simplifying Rational Expressions

Rational expressions can be simplified by factoring numerators and denominators and canceling common factors.

• Example: Simplify

  • Factor numerator:

  • Multiply numerators and denominators:

  • Simplify:

  • Note: and are negatives of each other, so

  • Final answer:

Simplifying Complex Rational Expressions

Complex rational expressions involve fractions within fractions. Simplify by finding a common denominator for the inner fractions, then simplify the overall expression.

• Example: Simplify

  • First, simplify

  • So,

  • Now, divide by :

  • Combine:

  • Factor

  • Final answer:

Mixture Problems

Solving Mixture Problems

Mixture problems involve combining solutions of different concentrations to achieve a desired concentration. Set up an equation based on the amount of pure substance in each solution.

• Key Formula: Amount of pure substance = volume × concentration

• Example: How many liters of pure water should be mixed with a 5-L solution of 80% acid to produce a mixture that is 70% water?

  • Let = liters of pure water to add.

  • 5 L of 80% acid = 20% water, so water in original solution: L

  • After adding liters of water, total water:

  • Total volume:

  • Set up equation:

  • Solve for :

  • L

Summary Table: Key Algebraic Properties