Radicals and Exponents

Evaluating and Simplifying Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. Simplifying these expressions often requires factoring and rationalizing denominators.

• Key Point: The principal square root of a number is its non-negative root. For example, is the non-negative number whose square is .

• Key Point: To simplify , factor each radicand and combine like terms:

Example:

• Combine:

Additional info: Always express answers in simplest radical form unless otherwise specified.

Evaluating Exponential Expressions

Exponential expressions involve repeated multiplication of a base. Negative and fractional exponents represent roots and reciprocals.

• Key Point:

• Example:

Solving Equations

Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one.

• Key Point: To solve , divide both sides by 3: .

Quadratic Equations

Quadratic equations are equations of the form . They can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic Formula:

• Example: For , , , :

• Discriminant:

• Solutions: , so or

Additional info: If the discriminant is negative, there are no real solutions.

Equations Involving Radicals

To solve equations involving square roots, isolate the radical and square both sides.

• Example:

• Example:

Factoring and Polynomials

Factoring Polynomials

Factoring is the process of writing a polynomial as a product of its factors.

• Key Point: To factor , look for two numbers that multiply to and add to .

• Example:

Multiplying Polynomials

Use the distributive property (FOIL for binomials) to multiply polynomials.

• Example:

Rational Expressions and Complex Fractions

Simplifying Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Simplify by factoring and reducing common factors.

• Example:

• Factor denominators: ,

• Find common denominator and combine.

Complex Fractions

Complex fractions have fractions in the numerator, denominator, or both. Simplify by finding a common denominator.

• Example:

• Rewrite denominator:

• So,

Functions and Applications

Function Notation and Applications

Functions describe relationships between variables. Applications often involve interpreting or solving for variables in context.

• Example: The annual revenue of a company is given by .

• To find when , set and solve for .

Solving for a Variable

To solve for a variable in terms of others, isolate the desired variable using algebraic operations.

• Example:

• Rearrange:

Inequalities and Number Lines

Graphing Solutions on a Number Line

Inequalities can be represented graphically on a number line, showing all possible solutions.

• Example: is shown as an arrow starting at 1 and extending to the right.

Additional Topics

Solving for y in Terms of x

Some equations require solving for one variable in terms of another, often involving radicals.

• Example:

• To solve for , rearrange and solve the resulting quadratic equation.

Word Problems and Applications

Word problems require translating real-world scenarios into algebraic equations and solving for unknowns.

• Example: Health care expenditures , find when .

• Solve for .