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Section 10.1 Evaluating Radical Expressions and Graphing
Square Root and Cube Root Functions
The number r is an nth root of x if denotes the principal nth root of x x is the _______________. is the _______________ symbol, or the radical. The natural number n is the _______________ or the order of the radical.
10.1 Lecture Guide: Evaluating Radical Expressions and Graphing Square Root and Cube Root Functions
.nr xn x
Radical Notation
Principal Square Root For any real number x > 0
Algebraically Verbally Numerically Graphically
if
and
denotes the principal square root of x.
y x
0.y
x
1,10,1 by 1,5,1
2y x
Algebraically Verbally Numerically Graphically
if y is a real number and
denotes the principal cube root of x.
Principal Cube Root For any real number x:
3y x
3 .y x
3 x
27, 27,5 by 3, 3, 1
Objective 1: Interpret and use radical notation.
Evaluate each expression. Use a calculator only to checkyour answer.
1. 2.81 3 27
Evaluate each expression. Use a calculator only to checkyour answer.
3. 4.4 81 3 27
Evaluate each expression. Use a calculator only to checkyour answer.
5. 6.4 16 3 125
Evaluate each expression. Use a calculator only to check your answer.
7. 9 16
Evaluate each expression. Use a calculator only to check your answer.
8. 9 16
Evaluate each expression. Use a calculator only to check your answer.
9. 9
16
Evaluate each expression. Use a calculator only to check your answer.
10. 31
27
Objective 2: Graph and analyze square root and cube root functions.
11. 4f x x
Type:
x-intercept:
y-intercept:
Domain:
Range:
Table
4
3
0
5
x y
Graph
-10
10
-10 10
y
x
12.
Type:
x-intercept:
y-intercept:
Domain:
Range:
Table 3 1f x x
9
2
1
0
7
x y
Graph
-10
10
-10 10
y
x
The domain of a real-valued function is the set of
all real numbers for which is also a real number. This
means the domain excludes all values that cause division by
__________________ or cause a ___________________
number under a square root symbol.
f x
Domain of a Function y f x
y f x
13. Determine the domain of 2 5f x x
using a calculator. Then graph
without
1 2 5y x
a window of
using
5,5,1 by 5,5,1 to verify your result.
14. Determine the domain of
using a calculator. Then graph
without
a window of
using
5,5,1 by 5,5,1 to verify your result.
5 2f x x
1 5 2y x
Determine the domain of the following. Use a graphing calculator only as a check on your work.
15. 3 4 5f x x
Determine the domain of the following. Use a graphing calculator only as a check on your work.
16. 3
xf x
x
Determine the domain of the following. Use a graphing calculator only as a check on your work.
17. 4 5f x x
Determine the domain of the following. Use a graphing calculator only as a check on your work.
18. 22 1f x x
Determine the domain of the following. Use a graphing calculator only as a check on your work.
19. 2 3f x x
Determine the domain of the following. Use a graphing calculator only as a check on your work.
20. 2 3f x x
Match each function with the appropriate graph. Hint: Completethese problems by recognizing the shape of the graph from the defining equation.
21.
______
22.
______
23.
______
24.
______
25.
______
26.
______
3 4 5f x x 22 1f x x
3x
f xx
2 3f x x
4 5f x x 2 3f x x
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
A. B. C.
D. E. F.
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
21. ______ 3 4 5f x x
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
A. B. C.
D. E. F.
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
22. ______ 22 1f x x
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
A. B. C.
D. E. F.
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
23. ______ 3x
f xx
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
A. B. C.
D. E. F.
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
24. ______ 2 3f x x
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
A. B. C.
D. E. F.
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
25. ______ 4 5f x x
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
A. B. C.
D. E. F.
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
-8 -4 4 8
-8
-4
4
8
x
y
26. ______ 2 3f x x
Mentally estimate the value of each expression and then use a calculator or a spreadsheet to approximate each value to the nearest thousandth.
27. 28.67 3 67
Mentally estimate the value of each expression and then use a calculator or a spreadsheet to approximate each value to the nearest thousandth.
29. 30.4 17 3 1 8
31. Determine to the nearest tenth of a centimeter the length of the hypotenuse of a right triangle with legs of 5 cm and 7 cm.
Use the formula 2 2 .c a b
32. Determine to the nearest tenth of a centimeter the radius of a sphere with a volume of Use the formula 3400 cm .
33
.4V
r
