
# 5.E: Radical Functions and Equations (Exercises)

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Exercise $$\PageIndex{1}$$

Simplify.

1. $$- \sqrt { 121 }$$
2. $$\sqrt { ( - 7 ) ^ { 2 } }$$
3. $$\sqrt { ( x y ) ^ { 2 } }$$
4. $$\sqrt { ( 6 x - 7 ) ^ { 2 } }$$
5. $$\sqrt [ 3 ] { 125 }$$
6. $$\sqrt [ 3 ] { - 27 }$$
7. $$\sqrt [ 3 ] { ( x y ) ^ { 3 } }$$
8. $$\sqrt [ 3 ] { ( 6 x + 1 ) ^ { 3 } }$$
9. Given $$f ( x ) = \sqrt { x + 10 }$$, find $$f(-1)$$ and $$f(6)$$.
10. Given $$g(x) = \sqrt [ 3 ] { x - 5 }$$, find $$g(4)$$ and $$g(13)$$.
11. Determine the domain of the function defined by $$g ( x ) = \sqrt { 5 x + 2 }$$.
12. Determine the domain of the function defined by $$g ( x ) = \sqrt [ 3 ] { 3 x - 1 }$$.

1. $$-11$$

3. $$|xy|$$

5. $$5$$

7. $$xy$$

9. $$f ( - 1 ) = 3 ; f ( 6 ) = 4$$

11. $$\left[ - \frac { 2 } { 5 } , \infty \right)$$

Exercise $$\PageIndex{2}$$

Simplify.

1. $$\sqrt [ 3 ] { 250 }$$
2. $$4 \sqrt [ 3 ] { 120 }$$
3. $$- 3 \sqrt [ 3 ] { 108 }$$
4. $$10 \sqrt [ 5 ] { \frac { 1 } { 32 } }$$
5. $$- 6 \sqrt [ 4 ] { \frac { 81 } { 16 } }$$
6. $$\sqrt [ 6 ] { 128 }$$
7. $$\sqrt [ 5 ] { - 192 }$$
8. $$- 3 \sqrt { 420 }$$

1. $$5 \sqrt [ 3 ] { 2 }$$

3. $$- 9 \sqrt [ 3 ] { 4 }$$

5. $$-9$$

7. $$- 2 \sqrt [ 5 ] { 6 }$$

Exercise $$\PageIndex{3}$$

Simplify.

1. $$\sqrt { 20 x ^ { 4 } y ^ { 3 } }$$
2. $$- 4 \sqrt { 54 x ^ { 6 } y ^ { 3 } }$$
3. $$\sqrt { x ^ { 2 } - 14 x + 49 }$$
4. $$\sqrt { ( x - 8 ) ^ { 4 } }$$

1. $$2 x ^ { 2 } | y | \sqrt { 5 y }$$

3. $$| x - 7 |$$

Exercise $$\PageIndex{4}$$

Simplify. (Assume all variable expressions are nonzero.)

1. $$\sqrt { 100 x ^ { 2 } y ^ { 4 } }$$
2. $$\sqrt { 36 a ^ { 6 } b ^ { 2 } }$$
3. $$\sqrt { \frac { 8 a ^ { 2 } } { b ^ { 4 } } }$$
4. $$\sqrt { \frac { 72 x ^ { 4 } y } { z ^ { 6 } } }$$
5. $$10 x \sqrt { 150 x ^ { 7 } y ^ { 4 } }$$
6. $$- 5 n ^ { 2 } \sqrt { 25 m ^ { 10 } n ^ { 6 } }$$
7. $$\sqrt [ 3 ] { 48 x ^ { 6 } y ^ { 3 } z ^ { 2 } }$$
8. $$\sqrt [ 3 ] { 270 a ^ { 10 } b ^ { 8 } c ^ { 3 } }$$
9. $$\sqrt [ 3 ] { \frac { a ^ { 3 } b ^ { 5 } } { 64 c ^ { 6 } } }$$
10. $$\sqrt [ 5 ] { \frac { a ^ { 26 } } { 32 b ^ { 5 } c ^ { 10 } } }$$
11. The period $$T$$ in seconds of a pendulum is given by the formula $$T = 2 \pi \sqrt { \frac { L } { 32 } }$$ where $$L$$ represents the length in feet of the pendulum. Calculate the period of a pendulum that is $$2 \frac{1}{2}$$ feet long. Give the exact answer and the approximate answer to the nearest hundredth of a second.
12. The time in seconds an object is in free fall is given by the formula $$t = \frac { \sqrt { s } } { 4 }$$ where $$s$$ represents the distance in feet the object has fallen. How long does it take an object to fall $$28$$ feet? Give the exact answer and the approximate answer to the nearest tenth of a second.
13. Find the distance between $$(−5, 6)$$ and $$(−3,−4)$$.
14. Find the distance between $$\left( \frac { 2 } { 3 } , - \frac { 1 } { 2 } \right)$$ and $$\left( 1 , - \frac { 3 } { 4 } \right)$$.

1. $$10 x y ^ { 2 }$$

3. $$\frac { 2 a \sqrt { 2 } } { b ^ { 2 } }$$

5. $$50 x ^ { 4 } y ^ { 2 } \sqrt { 6 x }$$

7. $$2 x ^ { 2 } y \sqrt [ 3 ] { 6 z ^ { 2 } }$$

9. $$\frac { a b \sqrt [ 3 ] { b ^ { 2 } } } { 4 c ^ { 2 } }$$

11. $$\frac { \pi \sqrt { 5 } } { 4 }$$ seconds; $$1.76$$ seconds

13. $$2 \sqrt { 26 }$$ units

Exercise $$\PageIndex{5}$$

Determine whether or not the three points form a right triangle. Use the Pythagorean theorem to justify your answer.

1. $$( - 4,5 ) , ( - 3 , - 1 ) , \text { and } ( 3,0 )$$
2. $$( - 1 , - 1 ) , ( 1,3 ) , \text { and } ( - 6,1 )$$

1. Right triangle

Exercise $$\PageIndex{6}$$

Simplify. Assume all radicands containing variables are nonnegative.

1. $$7 \sqrt { 2 } + 5 \sqrt { 2 }$$
2. $$8 \sqrt { 15 } - 2 \sqrt { 15 }$$
3. $$14 \sqrt { 3 } + 5 \sqrt { 2 } - 5 \sqrt { 3 } - 6 \sqrt { 2 }$$
4. $$22 \sqrt { a b } - 5 a \sqrt { b } + 7 \sqrt { a b } - 2 a \sqrt { b }$$
5. $$7 \sqrt { x } - ( 3 \sqrt { x } + 2 \sqrt { y } )$$
6. $$( 8 y \sqrt { x } - 7 x \sqrt { y } ) - ( 5 x \sqrt { y } - 12 y \sqrt { x } )$$
7. $$( 3 \sqrt { 5 } + 2 \sqrt { 6 } ) + ( 8 \sqrt { 5 } - 3 \sqrt { 6 } )$$
8. $$( 4 \sqrt [ 3 ] { 3 } - \sqrt [ 3 ] { 12 } ) - ( 5 \sqrt [ 3 ] { 3 } - 2 \sqrt [ 3 ] { 12 } )$$
9. $$( 2 - \sqrt { 10 x } + 3 \sqrt { y } ) - ( 1 + 2 \sqrt { 10 x } - 6 \sqrt { y } )$$
10. $$\left( 3 a \sqrt [ 3 ] { a b ^ { 2 } } + 6 \sqrt [ 3 ] { a ^ { 2 } b } \right) + \left( 9 a \sqrt [ 3 ] { a b ^ { 2 } } - 12 \sqrt [ 3 ] { a ^ { 2 } b } \right)$$
11. $$\sqrt { 45 } + \sqrt { 12 } - \sqrt { 20 } - \sqrt { 75 }$$
12. $$\sqrt { 24 } - \sqrt { 32 } + \sqrt { 54 } - 2 \sqrt { 32 }$$
13. $$2 \sqrt { 3 x ^ { 2 } } + \sqrt { 45 x } - x \sqrt { 27 } + \sqrt { 20 x }$$
14. $$5 \sqrt { 6 a ^ { 2 } b } + \sqrt { 8 a ^ { 2 } b ^ { 2 } } - 2 \sqrt { 24 a ^ { 2 } b } - a \sqrt { 18 b ^ { 2 } }$$
15. $$5 y \sqrt { 4 x ^ { 2 } y } - \left( x \sqrt { 16 y ^ { 3 } } - 2 \sqrt { 9 x ^ { 2 } y ^ { 3 } } \right)$$
16. $$\left( 2 b \sqrt { 9 a ^ { 2 } c } - 3 a \sqrt { 16 b ^ { 2 } c } \right) - \left( \sqrt { 64 a ^ { 2 } b ^ { 2 } c } - 9 b \sqrt { a ^ { 2 } c } \right)$$
17. $$\sqrt [ 3 ] { 216 x } - \sqrt [ 3 ] { 125 x y } - \sqrt [ 3 ] { 8 x }$$
18. $$\sqrt [ 3 ] { 128 x ^ { 3 } } - 2 x \sqrt [ 3 ] { 54 } + 3 \sqrt [ 3 ] { 2 x ^ { 3 } }$$
19. $$\sqrt [ 3 ] { 8 x ^ { 3 } y } - 2 x \sqrt [ 3 ] { 8 y } + \sqrt [ 3 ] { 27 x ^ { 3 } y } + x \sqrt [ 3 ] { y }$$
20. $$\sqrt [ 3 ] { 27 a ^ { 3 } b } - 3 \sqrt [ 3 ] { 8 a b ^ { 3 } } + a \sqrt [ 3 ] { 64 b } - b \sqrt [ 3 ] { a }$$
21. Calculate the perimeter of the triangle formed by the following set of vertices: $$\{ ( - 3 , - 2 ) , ( - 1,1 ) , ( 1 , - 2 ) \}$$.
22. Calculate the perimeter of the triangle formed by the following set of vertices: $$\{ ( 0 , - 4 ) , ( 2,0 ) , ( - 3,0 ) \}$$.

1. $$12 \sqrt { 2 }$$

3. $$9 \sqrt { 3 } - \sqrt { 2 }$$

5. $$4 \sqrt { x } - 2 \sqrt { y }$$

7. $$11 \sqrt { 5 } - \sqrt { 6 }$$

9. $$1 - 3 \sqrt { 10 x } + 9 \sqrt { y }$$

11. $$\sqrt { 5 } - 3 \sqrt { 3 }$$

13. $$- x \sqrt { 3 } + 5 \sqrt { 5 x }$$

15. $$12 x y \sqrt { y }$$

17. $$4 \sqrt [ 3 ] { x } - 5 \sqrt [ 3 ] { x y }$$

19. $$2 x \sqrt [ 3 ] { y }$$

21. $$4 + 2 \sqrt { 13 }$$ units

Exercise $$\PageIndex{7}$$

Multiply.

1. $$\sqrt { 6 } \cdot \sqrt { 15 }$$
2. $$( 4 \sqrt { 2 } ) ^ { 2 }$$
3. $$\sqrt { 2 } ( \sqrt { 2 } - \sqrt { 10 } )$$
4. $$( \sqrt { 5 } - \sqrt { 6 } ) ^ { 2 }$$
5. $$( 5 - \sqrt { 3 } ) ( 5 + \sqrt { 3 } )$$
6. $$( 2 \sqrt { 6 } + \sqrt { 3 } ) ( \sqrt { 2 } - 5 \sqrt { 3 } )$$
7. $$( \sqrt { a } - 5 \sqrt { b } ) ^ { 2 }$$
8. $$3 \sqrt { x y } ( \sqrt { x } - 2 \sqrt { y } )$$
9. $$\sqrt [ 3 ] { 3 a ^ { 2 } } \cdot \sqrt [ 3 ] { 18 a }$$
10. $$\sqrt [ 3 ] { 49 a ^ { 2 } b } \cdot \sqrt [ 3 ] { 7 a ^ { 2 } b ^ { 2 } }$$

1. $$3 \sqrt { 10 }$$

3. $$2 - 2 \sqrt { 5 }$$

5. $$22$$

7. $$a - 10 \sqrt { a b } + 25 b$$

9. $$3 a \sqrt [ 3 ] { 2 }$$

Exercise $$\PageIndex{8}$$

Divide. Assume all variables represent nonzero numbers and rationalize the denominator where appropriate.

1. $$\frac { \sqrt { 72 } } { \sqrt { 9 } }$$
2. $$\frac { 10 \sqrt { 48 } } { \sqrt { 64 } }$$
3. $$\frac { 5 } { \sqrt { 5 } }$$
4. $$\frac { \sqrt { 15 } } { \sqrt { 2 } }$$
5. $$\frac { 3 } { 2 \sqrt { 6 } }$$
6. $$\frac { 2 + \sqrt { 5 } } { \sqrt { 10 } }$$
7. $$\frac { 18 } { \sqrt { 3 x } }$$
8. $$\frac { 2 \sqrt { 3 x } } { \sqrt { 6 x y } }$$
9. $$\frac { 1 } { \sqrt [ 3 ] { 3 x ^ { 2 } } }$$
10. $$\frac { 5 a b ^ { 2 } } { \sqrt [ 3 ] { 5 a ^ { 2 } b } }$$
11. $$\sqrt [ 3 ] { \frac { 5 x z ^ { 2 } } { 49 x ^ { 2 } y ^ { 2 } z } }$$
12. $$\frac { 1 } { \sqrt [ 5 ] { 8 x ^ { 4 } y ^ { 2 } z } }$$
13. $$\frac { 9 x ^ { 2 } y } { \sqrt [ 5 ] { 81 x y ^ { 2 } z ^ { 3 } } }$$
14. $$\sqrt [ 5 ] { \frac { 27 a b ^ { 3 } } { 15 a ^ { 4 } b c ^ { 2 } } }$$
15. $$\frac { 1 } { \sqrt { 5 } - \sqrt { 3 } }$$
16. $$\frac { \sqrt { 3 } } { \sqrt { 2 } + 1 }$$
17. $$\frac { - 3 \sqrt { 6 } } { 2 - \sqrt { 10 } }$$
18. $$\frac { \sqrt { x y } } { \sqrt { x } - \sqrt { y } }$$
19. $$\frac { \sqrt { 2 } - \sqrt { 6 } } { \sqrt { 2 } + \sqrt { 6 } }$$
20. $$\frac { \sqrt { a } + \sqrt { b } } { \sqrt { a } - \sqrt { b } }$$
21. The base of a triangle measures $$2 \sqrt{6}$$ units and the height measures $$3 \sqrt{15}$$ units. Find the area of the triangle.
22. If each side of a square measures $$5+2 \sqrt{10}$$ units, find the area of the square.

1. $$2 \sqrt { 2 }$$

3. $$\sqrt { 5 }$$

5. $$\frac { \sqrt { 6 } } { 4 }$$

7. $$\frac { 6 \sqrt { 3 x } } { x }$$

9. $$\frac { \sqrt [ 3 ] { 9 x } } { 3 x }$$

11. $$\frac { \sqrt [ 3 ] { 35 x ^ { 2 } y z } } { 7 x y }$$

13. $$\frac { 3 x y \sqrt [ 5 ] { 3 x ^ { 4 } y ^ { 3 } z ^ { 2 } } } { z }$$

15. $$\frac { \sqrt { 5 } + \sqrt { 3 } } { 2 }$$

17. $$\sqrt { 6 } + \sqrt { 15 }$$

19. $$- 2 + \sqrt { 3 }$$

21. $$9 \sqrt { 10 }$$ square units

Exercise $$\PageIndex{9}$$

1. $$11 ^ { 1 / 2 }$$
2. $$2 ^ { 2 / 3 }$$
3. $$x ^ { 3 / 5 }$$
4. $$a ^ { - 4 / 5 }$$

1. $$\sqrt { 11 }$$

3. $$\sqrt [ 5 ] { x ^ { 3 } }$$

Exercise $$\PageIndex{10}$$

Write as a radical and then simplify.

1. $$16 ^ { 1 / 2 }$$
2. $$72 ^ { 1 / 2 }$$
3. $$8 ^ { 2 / 3 }$$
4. $$32 ^ { 1 / 3 }$$
5. $$\left( \frac { 1 } { 9 } \right) ^ { 3 / 2 }$$
6. $$\left( \frac { 1 } { 216 } \right) ^ { - 1 / 3 }$$

1. $$4$$

3. $$4$$

5. $$\frac{1}{27}$$

Exercise $$\PageIndex{11}$$

Perform the operations and simplify. Leave answers in exponential form.

1. $$6 ^ { 1 / 2 } \cdot 6 ^ { 3 / 2 }$$
2. $$3 ^ { 1 / 3 } \cdot 3 ^ { 1 / 2 }$$
3. $$\frac { 6 ^ { 5 / 2 } } { 6 ^ { 3 / 2 } }$$
4. $$\frac { 4 ^ { 3 / 4 } } { 4 ^ { 1 / 4 } }$$
5. $$\left( 64 x ^ { 6 } y ^ { 2 } \right) ^ { 1 / 2 }$$
6. $$\left( 27 x ^ { 12 } y ^ { 6 } \right) ^ { 1 / 3 }$$
7. $$\left( \frac { a ^ { 4 / 3 } } { a ^ { 1 / 2 } } \right) ^ { 2 / 5 }$$
8. $$\left( \frac { 16 x ^ { 4 / 3 } } { y ^ { 2 } } \right) ^ { 1 / 2 }$$
9. $$\frac { 56 x ^ { 3 / 4 } y ^ { 3 / 2 } } { 14 x ^ { 1 / 2 } y ^ { 2 / 3 } }$$
10. $$\frac { \left( 4 a ^ { 4 } b ^ { 2 / 3 } c ^ { 4 / 3 } \right) ^ { 1 / 2 } } { 2 a ^ { 2 } b ^ { 1 / 6 } c ^ { 2 / 3 } }$$
11. $$\left( 9 x ^ { - 4 / 3 } y ^ { 1 / 3 } \right) ^ { - 3 / 2 }$$
12. $$\left( 16 x ^ { - 4 / 5 } y ^ { 1 / 2 } z ^ { - 2 / 3 } \right) ^ { - 3 / 4 }$$

1. $$36$$

3. $$6$$

5. $$8 x ^ { 3 } y$$

7. $$a ^ { 1 / 3 }$$

9. $$4 x ^ { 1 / 4 } y ^ { 5 / 6 }$$

11. $$\frac { x ^ { 2 } } { 27 y ^ { 1 / 2 } }$$

Exercise $$\PageIndex{12}$$

Perform the operations with mixed indices.

1. $$\sqrt { y } \cdot \sqrt [ 5 ] { y ^ { 2 } }$$
2. $$\sqrt [ 3 ] { y } \cdot \sqrt [ 5 ] { y ^ { 3 } }$$
3. $$\frac { \sqrt [ 3 ] { y ^ { 2 } } } { \sqrt [ 3 ] { y } }$$
4. $$\sqrt { \sqrt [ 3 ] { y ^ { 2 } } }$$

1. $$\sqrt [ 10 ] { y ^ { 9 } }$$

3. $$\sqrt [ 15 ] { y ^ { 7 } }$$

Exercise $$\PageIndex{13}$$

Solve.

1. $$2 \sqrt { x } + 3 = 13$$
2. $$\sqrt { 3 x - 2 } = 4$$
3. $$\sqrt { x - 5 } + 4 = 8$$
4. $$5 \sqrt { x + 3 } + 7 = 2$$
5. $$\sqrt { 4 x - 3 } = \sqrt { 2 x + 15 }$$
6. $$\sqrt { 8 x - 15 } = x$$
7. $$x - 1 = \sqrt { 13 - x }$$
8. $$\sqrt { 4 x - 3 } = 2 x - 3$$
9. $$\sqrt { x + 5 } = 5 - \sqrt { x }$$
10. $$\sqrt { x + 3 } = 3 \sqrt { x } - 1$$
11. $$\sqrt { 2 ( x + 1 ) } - \sqrt { x + 2 } = 1$$
12. $$\sqrt { 6 - x } + \sqrt { x - 2 } = 2$$
13. $$\sqrt { 3 x - 2 } + \sqrt { x - 1 } = 1$$
14. $$\sqrt { 9 - x } = \sqrt { x + 16 } - 1$$
15. $$\sqrt [ 3 ] { 4 x - 3 } = 2$$
16. $$\sqrt [ 3 ] { x - 8 } = - 1$$
17. $$\sqrt [ 3 ] { x ( 3 x + 10 ) } = 2$$
18. $$\sqrt [ 3 ] { 2 x ^ { 2 } - x } + 4 = 5$$
19. $$\sqrt [ 3 ] { 3 ( x + 4 ) ( x + 1 ) } = \sqrt [ 3 ] { 5 x + 37 }$$
20. $$\sqrt [ 3 ] { 3 x ^ { 2 } - 9 x + 24 } = \sqrt [ 3 ] { ( x + 2 ) ^ { 2 } }$$
21. $$y ^ { 1 / 2 } - 3 = 0$$
22. $$y ^ { 1 / 3 } + 3 = 0$$
23. $$( x - 5 ) ^ { 1 / 2 } - 2 = 0$$
24. $$( 2 x - 1 ) ^ { 1 / 3 } - 5 = 0$$
25. $$( x - 1 ) ^ { 1 / 2 } = x ^ { 1 / 2 } - 1$$
26. $$( x - 2 ) ^ { 1 / 2 } - ( x - 6 ) ^ { 1 / 2 } = 2$$
27. $$( x + 4 ) ^ { 1 / 2 } - ( 3 x ) ^ { 1 / 2 } = - 2$$
28. $$( 5 x + 6 ) ^ { 1 / 2 } = 3 - ( x + 3 ) ^ { 1 / 2 }$$
29. Solve for $$g : t = \sqrt { \frac { 2 s } { g } }$$.
30. Solve for $$x:y = \sqrt [ 3 ] { x + 4 } - 2$$,
31. The period in seconds of a pendulum is given by the formula $$T = 2 \pi \sqrt { \frac { L } { 32 } }$$ where $$L$$ represents the length in feet of the pendulum. Find the length of a pendulum that has a period of $$1 \frac{1}{2}$$ seconds. Find the exact answer and the approximate answer rounded off to the nearest tenth of a foot.
32. The outer radius of a spherical shell is given by the formula $$r = \sqrt [ 3 ] { \frac { 3 V } { 4 \pi } } + 2$$ where $$V$$ represents the inner volume in cubic centimeters. If the outer radius measures $$8$$ centimeters, find the inner volume of the sphere.
33. The speed of a vehicle before the brakes are applied can be estimated by the length of the skid marks left on the road. On dry pavement, the speed $$v$$ in miles per hour can be estimated by the formula $$v = 2 \sqrt { 6 d }$$, where $$d$$ represents the length of the skid marks in feet. Estimate the length of a skid mark if the vehicle is traveling $$30$$ miles per hour before the brakes are applied.
34. Find the real root of the function defined by $$f ( x ) = \sqrt [ 3 ] { x - 3 } + 2$$.

1. $$25$$

3. $$21$$

5. $$9$$

7. $$4$$

9. $$4$$

11. $$7$$

13. $$1$$

15. $$\frac{11}{4}$$

17. $$−4, \frac{2}{3}$$

19. $$−5, \frac{5}{3}$$

21. $$9$$

23. $$9$$

25. $$1$$

27. $$12$$

29. $$g = \frac { 2 s } { t ^ { 2 } }$$

31. $$\frac { 18 } { \pi ^ { 2 } }$$ feet; $$1.8$$ feet

33. $$37.5$$ feet

Exercise $$\PageIndex{14}$$

Write the complex number in standard form $$a+bi$$.

1. $$5 - \sqrt { - 16 }$$
2. $$- \sqrt { - 25 } - 6$$
3. $$\frac { 3 + \sqrt { - 8 } } { 10 }$$
4. $$\frac { \sqrt { - 12 } - 4 } { 6 }$$

1. $$5 - 4 i$$

3. $$\frac { 3 } { 10 } + \frac { \sqrt { 2 } } { 5 } i$$

Exercise $$\PageIndex{15}$$

Perform the operations.

1. $$( 6 - 12 i ) + ( 4 + 7 i )$$
2. $$( - 3 + 2 i ) - ( 6 - 4 i )$$
3. $$\left( \frac { 1 } { 2 } - i \right) - \left( \frac { 3 } { 4 } - \frac { 3 } { 2 } i \right)$$
4. $$\left( \frac { 5 } { 8 } - \frac { 1 } { 5 } i \right) + \left( \frac { 3 } { 2 } - \frac { 2 } { 3 } i \right)$$
5. $$( 5 - 2 i ) - ( 6 - 7 i ) + ( 4 - 4 i )$$
6. $$( 10 - 3 i ) + ( 20 + 5 i ) - ( 30 - 15 i )$$
7. $$4 i ( 2 - 3 i )$$
8. $$( 2 + 3 i ) ( 5 - 2 i )$$
9. $$( 4 + i ) ^ { 2 }$$
10. $$( 8 - 3 i ) ^ { 2 }$$
11. $$( 3 + 2 i ) ( 3 - 2 i )$$
12. $$( - 1 + 5 i ) ( - 1 - 5 i )$$
13. $$\frac { 2 + 9 i } { 2 i }$$
14. $$\frac { i } { 1 - 2 i }$$
15. $$\frac { 4 + 5 i } { 2 - i }$$
16. $$\frac { 3 - 2 i } { 3 + 2 i }$$
17. $$10 - 5 ( 2 - 3 i ) ^ { 2 }$$
18. $$( 2 - 3 i ) ^ { 2 } - ( 2 - 3 i ) + 4$$
19. $$\left( \frac { 1 } { 1 - i } \right) ^ { 2 }$$
20. $$\left( \frac { 1 + 2 i } { 3 i } \right) ^ { 2 }$$
21. $$\sqrt { - 8 } ( \sqrt { 3 } - \sqrt { - 4 } )$$
22. $$( 1 - \sqrt { - 18 } ) ( 3 - \sqrt { - 2 } )$$
23. $$( \sqrt { - 5 } - \sqrt { - 10 } ) ^ { 2 }$$
24. $$( 1 - \sqrt { - 2 } ) ^ { 2 } - ( 1 + \sqrt { - 2 } ) ^ { 2 }$$
25. Show that both $$-5i$$ and $$5i$$ satisfy $$x^{2}+25=0$$.
26. Show that both $$1-2i$$ and $$1+2i$$ satisfy $$x^{2}-2x+5=0$$.

1. $$10 - 5 i$$

3. $$- \frac { 1 } { 4 } + \frac { 1 } { 2 } i$$

5. $$3+i$$

7. $$12+8i$$

9. $$15+8i$$

11. $$13$$

13. $$\frac{9}{2}-i$$

15. $$\frac { 3 } { 5 } + \frac { 14 } { 5 } i$$

17. $$35+60i$$

19. $$\frac{1}{2}i$$

21. $$4 \sqrt { 2 } + 2 i \sqrt { 6 }$$

23. $$- 15 + 10 \sqrt { 2 }$$

## Sample Exam

Exercise $$\PageIndex{16}$$

Simplify. (Assume all variables are positive.)

1. $$5 x \sqrt { 121 x ^ { 2 } y ^ { 4 } }$$
2. $$2 x y ^ { 2 } \sqrt [ 3 ] { - 64 x ^ { 6 } y ^ { 9 } }$$
3. Calculate the distance between $$(-5,-3)$$ and $$(-2,6)$$.
4. The time in seconds an object is in free fall is given by the formula $$t = \frac { \sqrt { s } } { 4 }$$ where $$s$$ represents the distance in feet that the object has fallen. If a stone is dropped into a $$36$$-foot pit, how long will it take to hit the bottom of the pit?

1. $$55 x ^ { 2 } y ^ { 2 }$$

3. $$3\sqrt{10}$$ units

Exercise $$\PageIndex{17}$$

Perform the operations and simplify. (Assume all variables are positive and rationalize the denominator where appropriate.)

1. $$\sqrt { 150 x y ^ { 2 } } - 2 \sqrt { 18 x ^ { 3 } } + y \sqrt { 24 x } + x \sqrt { 128 x }$$
2. $$3 \sqrt [ 3 ] { 16 x ^ { 3 } y ^ { 2 } } - \left( 2 x \sqrt [ 3 ] { 250 y ^ { 2 } } - \sqrt [ 3 ] { 54 x ^ { 3 } y ^ { 2 } } \right)$$
3. $$2 \sqrt { 2 } ( \sqrt { 2 } - 3 \sqrt { 6 } )$$
4. $$( \sqrt { 10 } - \sqrt { 5 } ) ^ { 2 }$$
5. $$\frac { \sqrt { 6 } } { \sqrt { 2 } + \sqrt { 3 } }$$
6. $$\frac { 2 x } { \sqrt { 2 x y } }$$
7. $$\frac { 1 } { \sqrt [ 5 ] { 8 x y ^ { 2 } z ^ { 4 } } }$$
8. Simplify: $$81 ^ { 3 / 4 }$$.
9. Express in radical form: $$x ^ { - 3 / 5 }$$.

1. $$7 y \sqrt { 6 x } + 2 x \sqrt { 2 x }$$

3. $$4 - 12 \sqrt { 3 }$$

5. $$- 2 \sqrt { 3 } + 3 \sqrt { 2 }$$

7. $$\frac { \sqrt [ 5 ] { 4 x ^ { 4 } y ^ { 3 } z } } { 2 x y z }$$

9. $$\frac { 1 } { \sqrt [ 5 ] { x ^ { 3 } } }$$

Exercise $$\PageIndex{18}$$

Solve.

1. $$\sqrt { x } - 5 = 1$$
2. $$\sqrt [ 3 ] { 5 x - 2 } + 6 = 4$$
3. $$5 \sqrt { 2 x + 5 } - 2 x = 11$$
4. $$\sqrt { 4 - 3 x } + 2 = x$$
5. $$\sqrt { 2 x + 5 } - \sqrt { x + 3 } = 2$$
6. The time in seconds an object is in free fall is given by the formula $$t = \frac { \sqrt { s } } { 4 }$$ where $$s$$ represents the distance in feet that the object has fallen. If a stone is dropped into a pit and it takes $$4$$ seconds to reach the bottom, how deep is the pit?
7. The width in inches of a container is given by the formula $$w = \frac { \sqrt [ 3 ] { 4 V } } { 2 } + 1$$ where $$V$$ represents the inside volume in cubic inches of the container. What is the inside volume of the container if the width is $$6$$ inches?

2. $$-\frac{6}{5}$$

4. $$\varnothing$$

6. $$256$$ feet

Exercise $$\PageIndex{19}$$

Perform the operations and write the answer in standard form.

1. $$\sqrt { - 3 } ( \sqrt { 6 } - \sqrt { - 3 } )$$
2. $$\frac { 4 + 3 i } { 2 - i }$$
3. $$6 - 3 ( 2 - 3 i ) ^ { 2 }$$
1. $$3 + 3 i \sqrt { 2 }$$
3. $$21 + 36 i$$