Roots, Radicals, and Root Functions 7 Chapter Roots, Radicals, and Root Functions

Simplifying Radicals, the Distance Formula, and Circles 7.3 Simplifying Radicals, the Distance Formula, and Circles 1. Use the product rule for radicals. 2. Use the quotient rule for radicals. 3. Simplify radicals. 4. Simplify products and quotients of radicals with different indexes. 5. Use the Pythagorean theorem. 6. Use the distance formula. 7. Find an equation of a circle given its center and radius.

Use the product rule for radicals. Objective 1 Use the product rule for radicals.

Product Rule for Radicals

Classroom Example 1 Using the Product Rule Multiply. Assume that all variables represent positive real numbers. a. b.

The expression cannot be simplified using the product rule. Classroom Example 2 Using the Product Rule Multiply. Assume that all variables represent positive real numbers. a. b. The expression cannot be simplified using the product rule.

Use the quotient rule for radicals. Objective 2 Use the quotient rule for radicals.

Quotient Rule for Radicals

Using the Quotient Rule Classroom Example 3 Using the Quotient Rule Simplify. Assume that all variables represent positive real numbers. a. b. c.

Objective 3 Simplify radicals.

Conditions for a Simplified Radical 1. The radicand has no factor raised to a power greater than or equal to the index. 2. The radicand has no fractions. 3. No denominator contains a radical. 4. Exponents in the radicand and the index of the radical have greatest common factor 1.

Simplifying Roots of Numbers Classroom Example 4 Simplifying Roots of Numbers Simplify. a. b. c. Cannot be simplified further

Simplifying Roots of Numbers (cont.) Classroom Example 4 Simplifying Roots of Numbers (cont.) Simplify. d. e.

Simplifying Radicals Involving Variables Classroom Example 5 Simplifying Radicals Involving Variables Simplify. Assume that all variables represent positive real numbers. a. b.

Simplifying Radicals Involving Variables (cont.) Classroom Example 5 Simplifying Radicals Involving Variables (cont.) Simplify. Assume that all variables represent positive real numbers. c. d.

Simplifying Radicals Using Lesser Indexes Classroom Example 6 Simplifying Radicals Using Lesser Indexes Simplify. Assume that all variables represent positive real numbers. a. b.

Meaning of Radicals

Simplify products and quotients of radicals with different indexes. Objective 4 Simplify products and quotients of radicals with different indexes.

Multiplying Radicals with Different Indexes Classroom Example 7 Multiplying Radicals with Different Indexes Simplify. The indexes, 2 and 3, have a least common index of 6. Use rational exponents to write each radical as a sixth root.

Use the Pythagorean theorem. Objective 5 Use the Pythagorean theorem.

The Pythagorean Theorem

Using the Pythagorean Theorem Classroom Example 8 Using the Pythagorean Theorem Find the length of the unknown side in each triangle.

Using the Pythagorean Theorem (cont.) Classroom Example 8 Using the Pythagorean Theorem (cont.) Find the length of the unknown side in each triangle.

Use the distance formula. Objective 6 Use the distance formula.

Distance Formula

Using the Distance Formula Classroom Example 9 Using the Distance Formula Find the distance between each pair of points. a. (2, –1) and (5, 3)

Using the Distance Formula (cont.) Classroom Example 9 Using the Distance Formula (cont.) Find the distance between each pair of points. b. (–3, 2) and (0, 4)

Find an equation of a circle given its center and radius. Objective 7 Find an equation of a circle given its center and radius.

Find an equation of a circle given its center and radius. A circle is the set of all points in a plane that lie a fixed distance from a fixed point. The fixed point is the center, and the fixed distance is the radius. We use the distance formula to find an equation of a circle.

Finding an Equation of a Circle and Graphing It Classroom Example 10 Finding an Equation of a Circle and Graphing It Find an equation of the circle with radius 4 and center (0, 0), and graph it. If the point (x, y) is on the circle, then the distance from (x, y) to the center (0, 0) is the radius 4.

Finding an Equation of a Circle and Graphing It Classroom Example 11 Finding an Equation of a Circle and Graphing It Find an equation of the circle with center at (3, –2) and radius 4, and graph it. 4

Equation of a Circle (Center-Radius Form)

Using the Center-Radius Form of the Equation of a Circle Classroom Example 12 Using the Center-Radius Form of the Equation of a Circle Find an equation of the circle with center at (2, –1) and radius