1. 4.8 Radical Equations and Power Functions
Learn properties of rational exponents
Learn radical notation
Use radical functions and solve radical equations
Understand properties and graphs of power functions
Use power functions to model data
Graph power functions having integer exponents
Solve equations involving rational exponents
Use power regression to model data

2. Properties of Rational Exponents (1 of 4)
Let r and p be rational numbers. Assume that b is a nonzero real number and that each expression is a real number. Property and Example

3. Properties of Rational Exponents (2 of 4)

4. Properties of Rational Exponents (3 of 4)

5. Properties of Rational Exponents (4 of 4)

6. Example: Applying properties of exponents
Solution

7. Example: Writing radicals with rational exponents
Use positive rational exponents to write each expression.
Solution

8. Functions Involving Radicals
Expressions involving radicals are sometimes used to define radical functions. Two common radical functions are the square root function and the cube root function.

9. Example: Evaluating radical functions
Evaluate each function for the given value of x.

10. Equations Involving Radicals
When solving equations that contain square roots, it is common to square each side of an equation and check the results.

11. Example: Solving an equation containing a square root (1 of 2)

12. Example: Solving an equation containing a square root (2 of 2)
Substituting these values in the original equation shows that the value of −5 is an extraneous solution because it does not satisfy the given equation. Therefore, the only solution is 3.

13. Example: Squaring twice (1 of 2)
Some equations require squaring twice.
Solution

14. Example: Squaring twice (2 of 2)
Both solutions check, so the solution set is {−1, 3}.

15. Power Functions and Models
Power functions typically have rational exponents.
A special type of power function is a root function.
Examples of power functions include:

16. Power Functions

17. Example: Graphing power functions
Discuss the effect that b has on the graph of f for x ≥ 1.
Solution Larger values of b cause the graph of f to increase faster.

18. Example: Modeling wing size of a bird (1 of 3)
Heavier birds have larger wings with more surface area than do lighter birds. For some species of birds, this relationship can be modeled by
where w is the weight of the bird in kilograms, with 0.1 ≤ w ≤ 5, and S is the surface area of the wings in square meters. a. Approximate S(0.5) and interpret the result. b. What weight corresponds to a surface area of 0.25 square meter?

19. Example: Modeling wing size of a bird (2 of 3)
The wings of a bird that weighs 0.5 kilogram have a surface area of about square meter.

20. Example: Modeling wing size of a bird (3 of 3)
Since w must be positive, the wings of a 1.4−kilogram bird have a surface area of about 0.25 square meter.

21. Graphs of Power Function Having Integer Exponents

22. Example: Interpreting the graph of a power function (1 of 4)
a. Is function f a power function? b. If possible, evaluate f(0) and f(2). c. Give the domain and range of f.
f. Does f(x) = f(−x)? Is f an odd or even function?

23. Example: Interpreting the graph of a power function (2 of 4)

24. Example: Interpreting the graph of a power function (3 of 4)

25. Example: Interpreting the graph of a power function (4 of 4)

26. Equations Involving Rational Exponents
Equations sometimes have rational exponents. The next example demonstrates a basic technique that can be used to solve some of these types of equations.

27. Example: Solving an equation that has rational exponents
Round to the nearest hundredth, and give graphical support. Solution

28. Example: Solve an equation having negative exponents
Solution

29. Power Regression
Rather than visually fit a curve to data, we can use least−squares regression to fit the data. Least−squares regression was introduced previously. In the next example, we apply this technique to data from biology.

30. Example: Modeling the length of a bird’s wing (1 of 4)
The table lists the weight W and the wingspan L for birds of a particular species.
W (kilograms)
0.5
1.5
2.0
2.5
3.0
L (meters)
0.77
1.10
1.22
1.31
1.40
b. Approximate the wingspan for a bird weighing 3.2 kilograms.

31. Example: Modeling the length of a bird’s wing (2 of 4)
Solution a. Let x be the weight W and y be the length L. Enter the data and select power regression.

32. Example: Modeling the length of a bird’s wing (3 of 4)
The results shown yield:

33. Example: Modeling the length of a bird’s wing (4 of 4)
b. If a bird weighs 3.2 kilograms, this model predicts the wingspan to be