2. Chapter 4: Rational, Power, and Root Functions
4.1 Rational Functions and Graphs
4.2 More on Rational Functions and Graphs
4.3 Rational Equations, Inequalities, Models, and Applications
4.4 Functions Defined by Powers and Roots
4.5 Equations, Inequalities, and Applications Involving Root Functions

3. 4.5 Equations, Inequalities, and Applications Involving Root Functions
Note: This property does not say that the two
equations are equivalent. The new equation may
have more solutions than the original.
e.g.
Power Property
If P and Q are algebraic expressions, then every solution of the equation P = Q is also a solution of the equation Pn = Qn, for any positive integer n.

4. 4.5 Solving Equations Involving Root Functions
Solving an Equation Involving a Root Function
Isolate a term involving a root on one side of the equation.
Raise both sides of the equation to a power that will eliminate the radical or rational exponent.
Solve the resulting equation. (If a root is still present after Step 2, repeat Steps 1 and 2.)
Check each proposed solution in the original equation.

5. 4.5 Solving an Equation Involving Square Roots
Example Solve
Analytic Solution
Isolate the radical.
Square both sides.
Write in standard form and solve.

6. 4.5 Solving an Equation Involving Square Roots
These solutions must be checked in the original
equation.

7. 4.5 Solving an Equation Involving Square Roots
Graphical Solution The equation in the second step
of the analytic solution has the same solution set as
the original equation. Graph
and solve y1 = y2.
The only solution is at x = 2.

8. 4.5 Solving an Equation Involving Cube Roots
Example Solve
Solution

9. 4.5 Solving an Equation Involving Roots (Squaring Twice)
Example Solve
Solution
Isolate radical.
Square both sides.
Isolate radical.
Square both sides.
Write in standard quadratic form and solve.
A check shows that –1 and 3 are solutions of the original equation.

10. 4.5 Solving Inequalities Involving Rational Exponents
Example Solve the inequality
Solution The associated equation solution in the
previous example was
Let
Use the x-intercept method to solve this inequality and determine the interval where the graph lies below the x-axis. The solution is the interval

11. 4.5 Application: Solving a Cable Installation Problem
A company wishes to run a utility cable from point A on the
shore to an installation at point B on the island (see figure).
The island is 6 miles from shore. It costs $400 per mile to run
cable on land and $500 per mile underwater. Assume that the
cable starts at point A and runs along the shoreline, then
angles and runs underwater
to the island. Let x represent
the distance from C at which
the underwater portion of the
cable run begins, and the
distance between A and C be
9 miles.

12. 4.5 Application: Solving a Cable Installation Problem
What are the possible values of x in this problem?
Express the cost of laying the cable as a function of x.
Find the total cost if 3 miles of cable are on land.
Find the point at which the line should begin to angle in order to minimize the total cost. What is this total cost?
Solution
The value of x must be real where
Let k be the length underwater. Using the Pythagorean theorem,

13. 4.5 Application: Solving a Cable Installation Problem
The cost of running cable is price  miles. If C is the total cost (in dollars) of laying cable across land and underwater, then
If 3 miles of cable are on land, then 3 = 9 – x, giving
x = 6.

14. 4.5 Application: Solving a Cable Installation Problem
Using the graphing calculator, find the minimum value of y1 = C(x) on the interval (0,9].
The minimum value of the function occurs when x = 8. So 9 – 8 = 1 mile should be along land, and
miles underwater. The cost is