Section 10.1 Radical Expressions and Functions Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives Radical Notation The Square Root Function The Cube Root Function

Radical Notation Every positive number a has two square roots, one positive and one negative. Recall that the positive square root is called the principal square root. The symbol is called the radical sign. The expression under the radical sign is called the radicand, and an expression containing a radical sign is called a radical expression. Examples of radical expressions:

Example Evaluate each square root. a. b. c.

Example Approximate to the nearest thousandth. Solution

Example Evaluate the cube root. a. b. c.

Example Find each root, if possible. a.b.c. Solution a. b. c. An even root of a negative number is not a real number.

Example Write each expression in terms of an absolute value. a.b.c. Solution a. b. c.

Example If possible, evaluate f(1) and f(  2) for each f(x). a.b. Solution a.b.

Example Calculate the hang time for a ball that is kicked 75 feet into the air. Does the hang time double when a ball is kicked twice as high? Use the formula Solution The hang time is The hang times is less than double.

Example Find the domain of each function. Write your answer in interval notation. a.b. Solution Solve 3 – 4x  0. The domain is b. Regardless of the value of x; the expression is always positive. The function is defined for all real numbers, and it domain is

Section 10.2 Rational Exponents Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives Basic Concepts Properties of Rational Exponents

Example Write each expression in radical notation. Then evaluate the expression and round to the nearest hundredth when appropriate. a.b.c. Solution a.b. c.

Example Write each expression in radical notation. Evaluate the expression by hand when possible. a.b. Solution a. b.

Example Write each expression in radical notation. Evaluate the expression by hand when possible. a.b. Solution a. Take the fourth root of 81 and then cube it. b. Take the fifth root of 14 and then fourth it. Cannot be evaluated by hand.

Example Write each expression in radical notation and then evaluate. a.b. Solution a.b.

Example Use rational exponents to write each radical expression. a. b. c. d.

Example Write each expression using rational exponents and simplify. Write the answer with a positive exponent. Assume that all variables are positive numbers. a.b.

Example (cont) Write each expression using rational exponents and simplify. Write the answer with a positive exponent. Assume that all variables are positive numbers. c.d.

Example Write each expression with positive rational exponents and simplify, if possible. a.b. Solution a. b.

Section 10.3 Simplifying Radical Expressions Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives Product Rule for Radical Expressions Quotient Rule for Radical Expressions

Product Rule for Radicals Consider the following example: Note: the product rule only works when the radicals have the same index.

Example Multiply each radical expression. a. b. c.

Example Multiply each radical expression. a. b. c.

Example Simplify each expression. a. b. c.

Example Simplify each expression. Assume that all variables are positive. a.b. c.

Example Simplify each expression. a.b.

Quotient Rule Consider the following examples of dividing radical expressions:

Example Simplify each radical expression. Assume that all variables are positive. a.b.

Example Simplify each radical expression. Assume that all variables are positive. a.b.

Example Simplify the radical expression. Assume that all variables are positive.

Example Simplify the expression. Solution

Example Simplify the expression. Solution

Section 10.4 Operations on Radical Expressions Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives Addition and Subtraction Multiplication Rationalizing the Denominator

Radicals Like radicals have the same index and the same radicand. LikeUnlike

Example If possible, add the expressions and simplify. a. b. c. d. The terms cannot be added because they are not like radicals. The expression contains unlike radicals and cannot be added.

Example Write each pair of terms as like radicals, if possible. a.b. Solution a.b.

Example Add the expressions and simplify. a.b. Solution a.b.

Example Simplify the expressions. a. b.

Example Subtract and simplify. Assume that all variables are positive. a.b.

Example Subtract and simplify. Assume that all variables are positive. a.b.

Example Multiply and simplify. Solution

Example Rationalize each denominator. Assume that all variables are positive. a.b.c. Solution a. b. c.

Conjugates

Example Rationalize the denominator of Solution

Example Rationalize the denominator of Solution

Example Rationalize the denominator of Solution

Section 10.6 Equations Involving Radical Expressions Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives Solving Radical Equations The Distance Formula Solving the Equation x n = k

If each side of an equation is raised to the same positive integer power, then any solutions to the given equations are among the solutions to the new equation. That is, the solutions to the equation a = b are among the solutions to a n = b n. POWER RULE FOR SOLVING EQUATIONS

Example Solve Check your solution. Solution Check: It checks.

Step 1: Isolate a radical term on one side of the equation. Step 2: Apply the power rule by raising each side of the equation to the power equal to the index of the isolated radical term. Step 3: Solve the equation. If it still contains radical, repeat Steps 1 and 2. Step 4: Check your answers by substituting each result in the given equation. SOLVING A RADICAL EQUATION

Example Solve Solution Step 1: To isolate the radical term, we add 3 to each side of the equation. Step 2: Square each side. Step 3: Solve the resulting equation.

Example (cont) Step 4: Check your answer by substituting into the given equation. Since this checks, the solution is x = −10.

Example Solve Check your results and then solve the equation graphically. Solution Symbolic Solution Check: It checks. Thus 1 is an extraneous solution.

Example (cont) Graphical Solution The solution 6 is supported graphically where the intersection is at (6, 4). The graphical solution does not give an extraneous solution.

Example Solve Solution The answer checks. The solution is 4.

Example Solve. Solution Step 1: The cube root is already isolated, so we proceed to Step 2. Step 2: Cube each side. Step 3: Solve the resulting equation.

Example (cont) Step 4: Check the answer by substituting into the given equation. Since this checks, the solution is x = 29.

Example Solve x 3/4 = 4 – x 2 graphically. This equation would be difficult to solve symbolically, but an approximate solution can be found graphically. Solution

Example A 6ft ladder is placed against a garage with its base 3 ft from the building. How high above the ground is the top of the ladder? Solution The ladder is 5.2 ft above ground.

The distance d between the points (x 1, y 1 ) and (x 2, y 2 ) in the xy-plane is DISTANCE FORMULA

Example Find the distance between the points (−1, 2) and (6, 4). Solution

Take the nth root of each side of x n = k to obtain 1. If n is odd, then and the equation becomes 2. If n is even and k > 0, then and the equation becomes (If k < 0, there are no real solutions.) SOLVING THE EQUATION x n = k

Example Solve each equation. a. x 3 = −216b. x 2 = 17 c. 3(x + 4) 4 = 48 Solution a. b. or

Example (cont) c. 3(x + 4) 4 = 48

Example The formula for the volume (V) of a sphere with a radius (r), is given by Solve for r. Solution

Section 10.7 Complex Numbers Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives Basic Concepts Addition, Subtraction, and Multiplication Powers of i Complex Conjugates and Division

PROPERTIES OF THE IMAGINARY UNIT i THE EXPRESSION

Example Write each square root using the imaginary i. a.b.c. Solution a. b. c.

Let a + bi and c + di be two complex numbers. Then (a + bi) + (c+ di) = (a + c) + (b + d)i Sum and (a + bi) − (c+ di) = (a − c) + (b − d)i. Difference SUM OR DIFFERENCE OF COMPLEX NUMBERS

Example Write each sum or difference in standard form. a.(−8 + 2i) + (5 + 6i)b. 9i – (3 – 2i) Solution a. (−8 + 2i) + (5 + 6i) = (−8 + 5) + (2 + 6)I = −3 + 8i b. 9i – (3 – 2i) = 9i – 3 + 2i = – 3 + (9 + 2)I = – i

Example Write each product in standard form. a.(6 − 3i)(2 + 2i)b. (6 + 7i)(6 – 7i) Solution a.(6 − 3i)(2 + 2i) = (6)(2) + (6)(2i) – (2)(3i) – (3i)(2i) = i – 6i – 6i 2 = i – 6i – 6(−1) = i

Example (cont) Write each product in standard form. a.(6 − 3i)(2 + 2i)b. (6 + 7i)(6 – 7i) Solution b. (6 + 7i)(6 – 7i) = (6)(6) − (6)(7i) + (6)(7i) − (7i)(7i) = 36 − 42i + 42i − 49i 2 = 36 − 49i 2 = 36 − 49(−1) = 85

The value of i n can be found by dividing n (a positive integer) by 4. If the remainder is r, then i n = i r. Note that i 0 = 1, i 1 = i, i 2 = −1, and i 3 = −i. POWERS OF i

Example Evaluate each expression. a.i 25 b. i 7 c. i 44 Solution a. When 25 is divided by 4, the result is 6 with the remainder of 1. Thus i 25 = i 1 = i. b. When 7 is divided by 4, the result is 1 with the remainder of 3. Thus i 7 = i 3 = −i. c. When 44 is divided by 4, the result is 11 with the remainder of 0. Thus i 44 = i 0 = 1.

Example Write each quotient in standard form. a.b. Solution a.

Example (cont) b.