Radical Expressions and Graph (7.1) EXAMPLE #1: EXAMPLE #2: EXAMPLE #3: Find roots of numbers (Objective #1) Figure #1:

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1 Radical Expressions and Graph (7.1) Find roots of numbers EXAMPLE #1: Figure #1: Find principal (positive) roots EXAMPLE #2: Find n th roots of n th powers (Objective #3) EXAMPLE #3: Figure #2:

2 7.1 Radical Expressions and Graphs Find roots of numbers. Find roots of numbers. Find principal roots of numbers. Find principal roots of numbers. (Objective #3) Find n th roots of n th powers. (Objective #3) Find n th roots of n th powers.

3 Rational Exponents (7.2) Use exponential notation for n th roots EXAMPLE #1: Figure #1: Define and use expressions of the form a m/n EXAMPLE #2: Figure #2: EXAMPLE #3:

4 Convert between radicals and rational exponents (Objective #3) EXAMPLE #4: Figure #3: EXAMPLE #5: Use the rules for exponents with rational exponents (Objective #4) Figure #4: EXAMPLE #6:

5 7.2 Rational Exponents Use exponential notation for the n th root. Define and use expressions of the form a m/n. (Objective #3) Convert between radical and rational exponents. (Objective #3) Convert between radical and rational exponents. (Objective #4) Use rules for exponents with rational exponents. (Objective #4) Use rules for exponents with rational exponents.

6 Simplifying Radicals, the Distance Formula, and Circles (7.3) Use the product rule for radicals EXAMPLE #1: Figure #1: Use the quotient rule for radicals EXAMPLE #2: Figure #2: Simplify radicals (Objective #3) EXAMPLE #3:

7 7.3 Simplifying Radicals, the Distance Formula, and Circles Use product rule for radicals. Use product rule for radicals. Use quotient rule for radicals. Use quotient rule for radicals. (Objective #3) Simplify radicals. (Objective #3) Simplify radicals.

8 Adding and Subtracting Radical Expressions (7.4) Simplify radical expressions involving addition and subtraction EXAMPLE #1: Figure #1: EXAMPLE #2:

9 7.4 Adding and Subtracting Radical Expressions Simplify by adding or subtracting. Simplify by adding or subtracting. Simplify by adding or subtracting. Simplify by adding or subtracting. Simplify by adding or subtracting. Simplify by adding or subtracting.

Figure #1: EXAMPLE #1: (b) EXAMPLE #2: Rationalize denominators with one radical term To rationalize a square root in the denominator with one radical term,

10 Multiplying and Dividing Radical Expressions (7.5) Multiply radical expressions Multiply the numbers outside the radicals and multiply the numbers inside the radicals. Figure #1: EXAMPLE #1: (b) EXAMPLE #2: Rationalize denominators with one radical term To rationalize a square root in the denominator with one radical term, multiply the top and bottom by the square root on the bottom as follows: Figure #2: EXAMPLE #3: For higher roots, simplify then multiply top and bottom by a radical with the same index as the bottom so that the number inside has a power equals to the index as follows: (a) EXAMPLE #4: Figure #3:

Rationalize denominators with binomials involving radicals (Objective #3) EXAMPLE #5: Write radical quotient in lowest terms (Objective #4) To reduce radical

11 Rationalize denominators with binomials involving radicals (Objective #3) EXAMPLE #5: Write radical quotient in lowest terms (Objective #4) To reduce radical quotient in lowest terms divide each term on top by the rational number the bottom as shown or simply factor the top completely and cross out any common factors. EXAMPLE #6:

12 7.5 Multiplying and Dividing Radical Expressions Multiply radical expressions. Multiply radical expressions. Rationalize denominator with one radical. Rationalize denominator with one radical. (Objective #3) Rationalize denominator with binomials involving radicals. (Objective #4) Write radical quotients in lowest terms.

Only even radicals can have extraneous solutions) Figure #1: EXAMPLE

13 Solving Equations with Radical (7.6) Solve radical equations using the power rule EXAMPLE #1: (Note: Only even radicals can have extraneous solutions) Figure #1: EXAMPLE #2: Solve radical equations with indexes greater than 2 Follow the same steps from above. EXAMPLE #3: Figure #2:

14 7.6 Solving Equations with Radical Solve. Solve. Solve. Solve. Solve. Solve.

15 Complex Numbers (7.7) Simplify numbers of the form where b > 0 EXAMPLE #1: Figure #1: Recognize subsets of the complex numbers COMPLEX NUMBERS EXAMPLE #2: Identify the real part and imaginary part. Add and subtract complex numbers (Objective #3) EXAMPLE #3:

16 Multiply complex numbers (Objective #4) EXAMPLE #4: Figure #2: (b) Divide complex numbers (Objective #5) EXAMPLE #5: Remember: Simplify powers of (Objective #6) EXAMPLE #6: (Note: If power is negative apply the following rule: )

17 7.7 Complex Numbers Simplify. Identify real and imaginary parts. (Objective #3) Adding and Subtracting complex numbers. (Objective #4) Multiplying complex numbers. (Objective #5) Dividing complex numbers. (Objective #6) Simplify powers of i.

EXAMPLE #1: Solve. Figure #1: The Quadratic Formula (8.

18 The Square Root Property and Completing the Square (8.1) Solve quadratic equations using the square root property EXAMPLE #1: Solve. Figure #1: The Quadratic Formula (8.2) Solve quadratic equations using the Quadratic Formula EXAMPLE #2: Figure #2:

19 8.1 & 8.2 Square Root Property and Quadratic Formula Solve using square root property. Solve using square root property. Solve using square root property. Solve using quadratic formula. Solve using quadratic formula. Solve using quadratic formula.

A slope of a line is the ratio between the change in a vertical distance (rise) to the change in a horizontal

The Slope of a Line (2.2) Find the slope of a line given two points on the line (Objective #1) A slope of a line is the ratio between the change in a vertical distance (rise) to the change in a horizontal