Solving Radical Equations and Inequalities Algebra II January 24 & 25

Evaluate the following expressions. Warm - Up Evaluate the following expressions. 1. 2. Solution: 16 Solution: - 8

Radicals/Exponents What does it mean when you have a fractions as an exponent? Such as: What this stands for is: the number in the numerator is the power, and the number in the denominator is the radical power. So I could write this in another way like:

Write each statement in a different form than given. 1. 2. 3. 1. 2. 3.

What about negative exponents What about negative exponents? Remember negative exponent means your doing the inverse.

Write each statement in a different form than given. 1. 2. 3. 1. 2. 3.

Solve the following rational exponential equation: Rational Exponents Solve the following rational exponential equation: OPTION #1 Step 1: Convert from exponent to radical form: Step 2: eliminate the radical: Step 3: Simplify:

Solve the following rational exponential equation: Rational Exponents Solve the following rational exponential equation: OPTION #2 Step 1: Raise to the reciprocal power of the original power: Step 2: Simplify:

Solving Radical Equations A radical equation is an equation with one or more radicals that have variables in their radicand. Solving Radical Equations Steps Step 1 Isolate the radical on one side of the equation if necessary. Step 2 Raise each side of the equation to the same power to get rid of the radical. Step 3 Solve the equation and check your solution.

Solve a radical Equation Write original equation. Cube each side. Simplify. Subtract 7 from each side. x = 10 Divide each side by 2. Solution x = 10 Check.

Try These… 1. SOLUTION: x = 512 2. SOLUTION: x = -9 3.

Rational Exponent Example What is the solution of the equation Write original equation. Divide each side by 3. Raise each side to the power of 3/2. x = 64 Simplify. Solution x = 64 Check.

Solve an equation with a rational Exponent. Write original equation. Add 1 to each side. Raise each side to the power of 4/3. Apply exponent properties. x = 14 Solve the equation. Solution x = 14 Check.

Try These… 1. 2. 3. SOLUTION: x = 25 SOLUTION: x = 1 SOLUTION: x = 6

Solve an equation w/ an extraneous solution Write original equation. Square each side. FOIL the left side and simplify the right. Write in standard form. Factor. x = 7 or x = -2 Solve. x = 7 (The -2 is extraneous) Check.

Solve an equation with 2 radicals METHOD 1 Write original equation. Square each side. FOIL the left side and simplify the right. Isolate the radical. Divide both sides by 2 . Square each side again. Simplify. Write in standard form and factor. x = 2 or x = -1 Solve. x = -1 (The 2 is extraneous) Check.

Solve an equation with 2 radicals METHOD 2 Write original equation. Graph y1 = Graph y2 = Find the point of intersection! You will find that the ONLY point of intersection is (-1, 2). Therefore, -1 is the only solution of the equation.

Try These… Solve the equation. Check for extraneous solutions. 1. 2. 3. SOLUTION: x = 1 SOLUTION: x = 0, 4 SOLUTION: x = 3

Solve radical inequalities Use a graph to solve SOLUTION Step 1 ENTER the function and y = 3 into the graphing calculator. Step 2 GRAPH the functions from Step 1. Step 3 Identify the x-values for which the graph of lies above the graph of y = 3. SOLUTION: x > 14

Solve the following radical inequalities (try by hand) 1. 2. SOLUTION: x > 32 SOLUTION: x ≥ 16

Class Work p. 447 #3-23 odd p. 456 #5, 7, 13, 17, 23, 27, 37, 45