1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
§7.2 Radical Functions
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 7.1 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
MTH 55
Review §
Any QUESTIONS About
§7.1 → Cube & nth Roots
Any QUESTIONS About HomeWork
§7.1 → HW-24

3. Cube Root The CUBE root, c, of a Number a is written as:
The number c is the cube root of a, if the third power of c is a; that is; if c3 = a, then

4. Example  Cube Root of No.s
Find Cube Roots
a) b) c)
SOLUTION a)
As 0.2·0.2·0.2 = 0.008 b)
As (−13)(−13)(−13) = −2197
As 33 = 27 and 43 = 64, so (3/4)3 = 27/64 c)

5. Rational Exponents
Consider a1/2a1/2. If we still want to add exponents when multiplying, it must follow from the Exponent PRODUCT RULE that
a1/2a1/2 = a1/2 + 1/2, or a1
Recall  [SomeThing]·[SomeThing] = [SomeThing]2
This suggests that a1/2 is a square root of a.

6. Definition of a1/n
When a is NONnegative, n can be any natural number greater than 1. When a is negative, n must be odd.
Note that the denominator of the exponent becomes the index and the BASE becomes the RADICAND.

7. nth Roots
nth root: The number c is an nth root of a number a if cn = a.
The fourth root of a number a is the number c for which c4 = a. We write for the nth root. The number n is called the index (plural, indices). When the index is 2 (for a Square Root), the Index is omitted.

8. Evaluating a1/n Evaluate Each Expression 27 (a) 271/3 = = 3 64 (b)
= 3 64
(b)
641/2 =
= 8
625 4
(c)
–6251/4
= –
= –5
–625 4
(d)
(–625)1/4 =
is not a real number because the radicand,
–625, is negative and the index is even.

9. Caveat on Roots
CAUTION: Notice the difference between parts (c) and (d) in the last Example.
The radical in part (c) is the negative fourth root of a positive number, while the radical in part (d) is the principal fourth root of a negative number, which is NOT a real no.
625 4
(c)
–6251/4
= –
= –5
–625 4
is not a real number because the radicand,
–625, is negative and the index is even.
(d)
(–625)1/4 =

10. Radical Functions
Given PolyNomial, P, a RADICAL FUNCTION Takes this form:
Example  Given f(x) = Then find f(3).
SOLUTION
To find f(3), substitute 3 for x and simplify.

11. Example  Exponent to Radical
Write an equivalent expression using RADICAL notation
a) b) c)
SOLUTION a) c) b)

12. Example  Radical to Exponent
Write an equivalent expression using EXPONENT notation
a) b)
SOLUTION a) b)

13. Exponent ↔ Index Base ↔ Radicand
From the Previous Examples Notice:
The denominator of the exponent becomes the index. The base becomes the radicand.
The index becomes the denominator of the exponent. The radicand becomes the base.

14. Definition of am/n
For any natural numbers m and n (n not 1) and any real number a for which the radical exists,

15. Example  am/n Radicals
Rewrite as radicals, then simplify
a. 272/3 b. 2433/4 c. 95/2
SOLUTION

16. Example  am/n Exponents
Rewrite with rational exponents
SOLUTION

17. Definition of a−m/n
For any rational number m/n and any positive real number a the NEGATIVE rational exponent:
That is, am/n and a−m/n are reciprocals

18. Caveat on Negative Exponents
A negative exponent does not indicate that the expression in which it appears is negative; i.e.;

19. Example  Negative Exponents
Rewrite with positive exponents, & simplify
a. 8−2/3 b. 9−3/2x1/ c.
SOLUTION

20. Example  Speed of Sound
Many applications translate to radical equations.
For example, at a temperature of t degrees Fahrenheit, sound travels S feet per second According to the Formula

21. Example  Speed of Sound
During orchestra practice, the temperature of a room was 74 °F. How fast was the sound of the orchestra traveling through the room?
SOLUTION: Substitute 74 for t in the Formula and find an approximation using a calculator.

22. WhiteBoard Work Problems From §7.2 Exercise Set The MACH No. M
4, 10, 18, 32, 48, 54, 130
The MACH No. M

23. Ernst Mach Fluid Dynamicist
All Done for Today
Ernst Mach Fluid Dynamicist
Born 8Feb1838 in Brno, Austria
Died 19Feb1916 (aged 78) in Munich, Germany

24. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –

26. Graph y = |x|
Make T-table