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Numerical Roots and Radicals Chapter Questions1. What are the properties of a square?2. What does taking the square root have to do with the area of a square?3. Why is it helpful to memorize perfect squares?4. What can be helpful when finding the square roots of numbers greater than 400?5. Why is it helpful to memorize perfect squares?6. Explain how to take the square root of a fraction or a decimal.7. Explain how to approximate a square root.8. What is the difference between an irrational and rational number?9. Why would we simplify a non-perfect square root instead of just estimating it?10. How do you solve an equation with perfect square and cube roots?
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Numerical Roots and Radicals Chapter Problems
Squares, Square Roots & Perfect SquaresClasswork1. A square has an area of 9 units2.
a. What is the side length of a square of this area?
b. Draw a square with an area of 9 units2.
c. What is the square root of 9?
d. Explain why your answers in parts (a) and (c) are the same.
2. Fill in the following table:
Side Lengthof a square
(units)
Area ofthe square
(units2)
1
2
3
4
5
6
7
8
9
10
11
12
13
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3. Explain how the table above helps you find the square root of 121?
4. Simplify each square root.a. √25b. √64c. √81d. √49e. √16
Homework
5. A square has an area of 36 units2.a. What is the side length of a square of this area?
b. Draw a square with an area of 36 units2.
c. What is the square root of 36?
d. Explain why your answers in parts (a) and (c) are the same.
6. Fill in the following table:
Side Lengthof a square
(units)
Area of thesquare(units2)
14
15
16
17
18
19
20
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7. Simplify each square root.a. √289b. √400c. √196d. √361e. √144
Squares of Numbers Greater Than 20
Classwork
8. Fill in the following table:
Side Lengthof a square
(units)
Area of thesquare(units2)
10
20
30
40
50
60
70
80
90
100
9. If you compare that to the table of side lengths from 1-10, what pattern do you notice?
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10. Simplify each square root.a. √2809b. √7921c. √484d. √6400e. √2025f. √225g. √841h. √9409i. √961j. √4356
Homework
11. Simplify each square root.a. √5041b. √1296c. √8464d. √3025e. √3721f. √6889g. √576h. √2401i. √2500j. √289
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Simplifying Perfect Square Radical ExpressionsClasswork
12. Simplify each square root.a. √25b. √64c. −√81d. √−81e. √49f.
g.
h.
i.
j. −k. √. 64l. √. 0081m. −√. 25n. √. 0016o. √−.04
Homework
13. Simplify each square root.a. √289b. -√400c. √64d. √361e. √−10000f.
g.
h.
i. -
j.
k. √−.09l. −√. 0196m. √. 49n. √. 0361o. √. 25
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Approximating Square RootsClasswork14. What two integers do the following square roots fall between?
a.b.c.d.e.
15. Draw and label a number line from 0 to 10. Place the following square roots on the number line.a.b.c.d.e.
16. Estimate the following square roots.a.b.c.d.e.f.g.h.i.j.
17. Approximate the square root to the nearest integer
a.
b.
c.
d.
e.
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Approximating Square RootsClasswork14. What two integers do the following square roots fall between?
a.b.c.d.e.
15. Draw and label a number line from 0 to 10. Place the following square roots on the number line.a.b.c.d.e.
16. Estimate the following square roots.a.b.c.d.e.f.g.h.i.j.
17. Approximate the square root to the nearest integer
a.
b.
c.
d.
e.
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Approximating Square RootsClasswork14. What two integers do the following square roots fall between?
a.b.c.d.e.
15. Draw and label a number line from 0 to 10. Place the following square roots on the number line.a.b.c.d.e.
16. Estimate the following square roots.a.b.c.d.e.f.g.h.i.j.
17. Approximate the square root to the nearest integer
a.
b.
c.
d.
e.
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Homework18. What two integers do the following square roots fall between?
a.b.c.d.e.
19. Draw and label a number line from 0 to 10. Place the following square roots on the number line.a.b.c.d.e.
20. Estimate the following square roots.a.b.c.d.e.f.g.h.i.j.
21. Approximate the square root to the nearest integer
a.
b.
c.
d.
e.
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Homework18. What two integers do the following square roots fall between?
a.b.c.d.e.
19. Draw and label a number line from 0 to 10. Place the following square roots on the number line.a.b.c.d.e.
20. Estimate the following square roots.a.b.c.d.e.f.g.h.i.j.
21. Approximate the square root to the nearest integer
a.
b.
c.
d.
e.
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Homework18. What two integers do the following square roots fall between?
a.b.c.d.e.
19. Draw and label a number line from 0 to 10. Place the following square roots on the number line.a.b.c.d.e.
20. Estimate the following square roots.a.b.c.d.e.f.g.h.i.j.
21. Approximate the square root to the nearest integer
a.
b.
c.
d.
e.
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Rational & Irrational NumbersClasswork22. Circle the numbers below that are rational
a. 3.5b. √6c. πd.
e. √10f. −√49g. √108h. 0.25i.
j. 0.4Homework23. Circle the numbers below that are irrational.
a.
b. √7c. √81d. 6.75e.f. √121g. √61h. πi. √225j. 0.18
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Radical Expressions Containing Variables
Classwork24. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
Homework25. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
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Radical Expressions Containing Variables
Classwork24. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
Homework25. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
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Radical Expressions Containing Variables
Classwork24. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
Homework25. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
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Simplifying Non-Perfect Squre RadicandsClasswork26. Simplify
a.b.c.d.e.f.g.h.i.j.
k.
l.
m.
Homework27. Simplify
a.b.c.d.e.f.g.h.i.j.
k.
l.
m.
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Simplifying Non-Perfect Squre RadicandsClasswork26. Simplify
a.b.c.d.e.f.g.h.i.j.
k.
l.
m.
Homework27. Simplify
a.b.c.d.e.f.g.h.i.j.
k.
l.
m.
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Simplifying Non-Perfect Squre RadicandsClasswork26. Simplify
a.b.c.d.e.f.g.h.i.j.
k.
l.
m.
Homework27. Simplify
a.b.c.d.e.f.g.h.i.j.
k.
l.
m.
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Simplifying Roots of VariablesClasswork28. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
Homework29. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
i.
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Simplifying Roots of VariablesClasswork28. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
Homework29. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
i.
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Simplifying Roots of VariablesClasswork28. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
Homework29. Simplify
a.
b.
c.
d.
e.
f.
g.
h.
i.
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Properties of ExponentsClasswork30. Complete each equation for the missing value:
a. (52)(55) = 5?
b. (127)(123) = 12?
c. (3-2)(35) = 3?
d. (49)(4-3) = 4?
e. (54)(5?) = 512
f. (107)(10?)(10-6) = 103
g. 34 ÷ 32 = 3?
h. = 5?
i. = 9?
j. 124 ÷ 126 = 12?
k. 108 ÷ 10? = 103
l. = 24
Homework31. Complete each equation for the missing value:
a. (122)(127) = 12?
b. (25)(22) = 2?
c. (5-3)(55) = 5?
d. (158)(15-5) = 15?
e. (67)(6?) = 615
f. (11-6)(11?)(118) = 115
g. 77 ÷ 73 = 7?
h. = 11?
i. 37 ÷ 39 = 3?
6
9
5
5
8
5
9
9
3
?
2
2
6
10
11
11
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j. = 2?
k. = 132
l. 5? ÷ 56 = 53
Solving Equations with Perfect Square and Cube RootsClasswork32. Solve.
a. 4 = 32b. = 28c. = −25d. 6 = 864e. 3 = 147
Homework33. Solve.
a. 21 = −21b. = 125c. 7 = 252d. −6 = 162e. = 4
10
6
2
2
?
6
13
13
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Numerical roots & Radicals Multiple choice Questions
Determine whether the given numbers are perfect squares. Circle your answer.
1) 1 Yes No
2) 8 Yes No
3) 16 Yes No
4) 25 Yes No
5) 82 Yes No
Circle the simplified version of each square root:
6) √144a. 14b. 12c. 72d. 21
7)
a. 10b. 6c. 0.6d. 18
8) −√. 0049a. -7b. 0.7c. 0.07d. -0.07
Circle whether the given number is rational or irrational9) π rational irrational10) 0.875 rational irrational11))√39 rational irrational
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Between what two integers do the following square roots fall?
12) √45a. 4 & 5b. 6 & 7c. 7 & 8d. 5 & 6
13)√125a. 11 & 12b. 12 & 13c. 13 & 14d. 14 & 15
14)Simplify: √45a. 40√5b. 2√5c. 3√5d. 9√5
15)(47)(43) = 4?
a. 10b. 24c. 4d. 5
Short Constructed Response – Write the correct answer for each question. No partial credit will begiven.
16) Approximate √47 ≈ ________
17) Approximate √230 ≈ ________
18) Solve: 5 2 = 180
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19) Solve:
20) 75 ___________________
21) ∶ 200 _____________________
22) Find the missing value 114 ÷ 116 = 11? ____________________
23) = ?________________
24) (67)(6-2) = 6?
25) Simplify √ ____________
1082
3
x
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Extended Constructed Response - Solve the problem, showing all work. Partial credit may be given.
26) Write two exponential expressions with like bases. Leave all answers in simplified exponential form.
a. Expression 1:
Expression 2:
b. Multiply your expressions.
c. Divide your expressions.
d. Raise your first expression to the 5th power.
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3 units
3 units
Answer Key
1.a. 3 unitsb.
c. 3d. Area of a Square = Side2 and 9 = 32
2.Side Lengthof a Square
(units)
Area of thesquare(units2)
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
11 121
12 144
13 169
3. Since Area of a square = side2, the square root of the area = side. So, √121 = 11.
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6 units
6 units
4.a. 5b. 8c. 9d. 7e. 4
5.a. 6 unitsb.
c. 6d. Area = Side2 and 36 = 62
6.Side Lengthof a square
(units)
Area of thesquare(units2)
14 196
15 225
16 256
17 289
18 324
19 361
20 400
7.a. 17b. 20c. 14d. 19e. 12
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8.Side Lengthof a square
(units)
Area of thesquare(units2)
10 100
20 400
30 900
40 1600
50 2500
60 3600
70 4900
80 6400
90 8100
100 10,000
9. Each answer in this table is 100 times greater than the corresponding answer in the other table. (or102 times greater).
10.a. 53b. 89c. 22d. 80e. 45f. 15g. 29h. 97i. 31j. 66
11.a. 71b. 36c. 92d. 55e. 61f. 83g. 24h. 49i. 50j. 17
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12.a. 5b. 8c. -9d. No real solutione. 7
f.
g. No real solutionh. ½
i.
j. – ½k. 0.8l. 0.09m. -0.5n. 0.04o. No real solution
13.a. 17b. -20c. 8d. 19e. No real solutionf. 1/3g. ½h. No real solutioni. -3/4j. 1/10k. No real solutionl. -0.14m. 0.7n. 0.19o. 0.5
14.a. 8 and 9b. 12 and 13c. 2 and 3d. 7 and 8e. 10 and 11
15.
5
7
5
7
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12.a. 5b. 8c. -9d. No real solutione. 7
f.
g. No real solutionh. ½
i.
j. – ½k. 0.8l. 0.09m. -0.5n. 0.04o. No real solution
13.a. 17b. -20c. 8d. 19e. No real solutionf. 1/3g. ½h. No real solutioni. -3/4j. 1/10k. No real solutionl. -0.14m. 0.7n. 0.19o. 0.5
14.a. 8 and 9b. 12 and 13c. 2 and 3d. 7 and 8e. 10 and 11
15.
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12.a. 5b. 8c. -9d. No real solutione. 7
f.
g. No real solutionh. ½
i.
j. – ½k. 0.8l. 0.09m. -0.5n. 0.04o. No real solution
13.a. 17b. -20c. 8d. 19e. No real solutionf. 1/3g. ½h. No real solutioni. -3/4j. 1/10k. No real solutionl. -0.14m. 0.7n. 0.19o. 0.5
14.a. 8 and 9b. 12 and 13c. 2 and 3d. 7 and 8e. 10 and 11
15.
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16.a. 2.45b. 8.37c. 7.42d. 3.74e. 10.29f. 6.4g. 8.94h. 8.06i. 2.83j. 15.26
17.a. 7b. 6c. 8d. 3e. 9
18.a. 12 and 13b. 3 and 4c. 9 and 10d. 8 and 9e. 13 and 14
19.
20.a. 8.83b. 2.65c. 7.94d. 5.39e. 6.48f. 11.75g. 17.32h. 12.17i. 4.58j. 7.21
21.a. 4b. 6c. 4d. 6e. 7
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16.a. 2.45b. 8.37c. 7.42d. 3.74e. 10.29f. 6.4g. 8.94h. 8.06i. 2.83j. 15.26
17.a. 7b. 6c. 8d. 3e. 9
18.a. 12 and 13b. 3 and 4c. 9 and 10d. 8 and 9e. 13 and 14
19.
20.a. 8.83b. 2.65c. 7.94d. 5.39e. 6.48f. 11.75g. 17.32h. 12.17i. 4.58j. 7.21
21.a. 4b. 6c. 4d. 6e. 7
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16.a. 2.45b. 8.37c. 7.42d. 3.74e. 10.29f. 6.4g. 8.94h. 8.06i. 2.83j. 15.26
17.a. 7b. 6c. 8d. 3e. 9
18.a. 12 and 13b. 3 and 4c. 9 and 10d. 8 and 9e. 13 and 14
19.
20.a. 8.83b. 2.65c. 7.94d. 5.39e. 6.48f. 11.75g. 17.32h. 12.17i. 4.58j. 7.21
21.a. 4b. 6c. 4d. 6e. 7
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22.a. Rationalb. Irrationalc. Irrationald. Rationale. Irrationalf. Rationalg. Irrationalh. Rationali. Rationalj. Rational
23.a. Rationalb. Irrationalc. Rationald. Rationale. Rationalf. Rationalg. Irrationalh. Irrationali. Rationalj. Rational
24.a.
b.
c.
d.e.f.
g.h.
25.a.
b.c.
d.e.f.
g.
h.
3b b3b
b b2b b2b
b4b4b b
3x x2x x
x2x4x3x
x x4x x
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26.a. 5b. 4
c. 2
d. 2
e. 5
f. 10
g. 5
h. 2
i. 3
j. 6
k. 105
l. 33
m. 6627.
a. 4
b.
c.
d.
e.
f.
g.
h.i.
j.
k.
l.m.
28.a.
b.
c.
d.
e.
2
3
5
2
3
3
5
7
7
2
3
2
6
6
5 6
4 5
2 6
10 5
6 3
2 30
7 3
2 21
8 5
56 5
120 3
112 2
2 23 2x y z x4 41x y z z
2 3 30x y z y3 4 47 x y z yz
33 3x yz yz
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f.
g.
h.
i.
j.
k.29.
a.
b.
c.
d.
e.
f.
g.
h.
i.30.
a. 7b. 10c. 3d. 6e. 8f. 2g. 2h. 3i. -3j. -2k. 5l. 7
31.a. 9b. 7c. 2d. 3e. 8f. 3
2 2 42 7x y z y
2 42 14x y z x
3 35x y z x
yzyx 3133
2 3 4 58x y z xz
4 2 17x y z y
2 4 22 10x y z xz3 3 3x y z yz
2 3 22 6x y z
3 2 3 65x y z y
2 14x y z x2 2 3 10x y z yz
yzyzx 72 22
4 22x y z z
4 33 3x y z xy
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g. 4h. 4i. -2j. -4k. 4l. 9
32.a. 2b. ±14c. -5d. ±12e. ±7
33.a. -1b. ±25c. ±6d. -3e. 4
Review answers1. Yes2. No3. Yes4. Yes5. No6. B7. C8. D9. Irrational
10. rational11. irrational12. b13. a14. c15. a16. 717. 1518. 6
19. -620. 5x7y√321. 10xyz2√222. -223. 424. 525. x5
26. Expressions will vary