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 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
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.
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
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.