Numerical Roots and Radicals Chapter Questions
Roots and Radicals Chapter Questions
1. 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. Explain how to take the square root of a fraction or a decimal.
6. Explain how to approximate a square root.
7. What is the difference between an irrational and rational number?
8. Explain how to convert between the different forms of a rational number.
Roots and Radicals Chapter Problems
Squares, Square Roots & Perfect Squares
Classwork
1. 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 Length of a |Area of the |
|square |square |
|(units) |(units2) |
|1 | |
|2 | |
|3 | |
|4 | |
|5 | |
|6 | |
|7 | |
|8 | |
|9 | |
|10 | |
|11 | |
|12 | |
|13 | |
3. Explain how the table above helps you find the square root of 121?
4. Simplify each square root.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
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 Length of a |Area of the square|
|square |(units2) |
|(units) | |
|14 | |
|15 | |
|16 | |
|17 | |
|18 | |
|19 | |
|20 | |
7. Simplify each square root.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
Squares of Numbers Greater Than 20
Classwork
8. Fill in the following table:
|Side Length of a |Area of the square|
|square |(units2) |
|(units) | |
|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?
10. Simplify each square root.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
g. [pic]
h. [pic]
i. [pic]
j. [pic]
Homework
11. Simplify each square root.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
g. [pic]
h. [pic]
i. [pic]
j. [pic]
Simplifying Perfect Square Radical Expressions
Classwork
12. Simplify each square root.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
g. [pic]
h. [pic]
i. [pic]
j. [pic]
k. [pic]
l. [pic]
m. [pic]
n. [pic]
o. [pic]
Homework
13. Simplify each square root.
a. [pic]
b. -[pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
a. [pic]
b. [pic]
c. -[pic]
d. [pic]
g. [pic]
h. [pic]
i. [pic]
j. [pic]
k. [pic]
Approximating Square Roots
Classwork
1. What two integers do the following square roots fall between?
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
2. Draw and label a number line from 0 to 10. Place the following square roots on the number line.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
3. Estimate the following square roots.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
g. [pic]
h. [pic]
i. [pic]
j. [pic]
4. Approximate the square root to the nearest integer
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
5. For what integer x is [pic] closest to 7.42?*
6. For what integer x is [pic] closest to 5.1?*
7. For what integer x is [pic] closest to 3.9?*
Homework
8. What two integers do the following square roots fall between?
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
9. Draw and label a number line from 0 to 10. Place the following square roots on the number line.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
10. Estimate the following square roots.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
g. [pic]
h. [pic]
i. [pic]
j. [pic]
11. Approximate the square root to the nearest integer
a.[pic]
b.[pic]
c.[pic]
d.[pic]
e. [pic]
12. For what integer x is [pic] closest to 5.2?*
13. For what integer x is [pic] closest to 6.1?*
14. For what integer x is [pic] closest to 6.9?*
Rational & Irrational Numbers
Classwork.
15. Circle the numbers below that are rational
a. 3.5
b. [pic]
c. π
d. [pic]
e. [pic]
f. [pic]
g. [pic]
h. 0.25
i. [pic]
j. 0.[pic]
16. Given the statement “If x is a rational number then [pic] is irrational.” Which values of x make the statement false?* A) 15 B) 24 C) 4 D) 20
17. Given the statement “If x is a rational number then [pic] is irrational.” Which values of x make the statement false?* A) 25 B) 49 C) 5 D) 59
Homework
18. Circle the numbers below that are irrational.
a. [pic]
b. [pic]
c. [pic]
d. 6.75
e. [pic]
f. [pic]
g. [pic]
h. π
i. [pic]
j. 0.[pic]
19. Given the statement “If x is a rational number then [pic] is irrational.” Which values of x make the statement false?* A) 36 B) 8 C) 16 D) 121
20. Given the statement “If x is a rational number then [pic] is irrational.” Which values of x make the statement false?* A) 144 B) 12 C) 48 D)30
Converting Repeating Decimals to Fractions and Decimal Expansion
Classwork
21. Write each repeating decimal as a fraction in simplest form*
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
22. Find the decimal expansion of the following*:
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
Homework
23. Write each repeating decimal as a fraction in simplest form*
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
24. Find the decimal expansion of the following*:
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
Properties of Exponents
Classwork
25. 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. [pic]= 5?
i. [pic] = 9?
j. 124 ÷ 126 = 12?
k. 108 ÷ 10? = 103
l. [pic]= 24
26. A rectangle has a length of [pic]mm and a width of [pic]mm. Write an expression for the area of the rectangle as a power of 5.*
27. Express the volume of a cube with a side length of [pic]inches as a power of 7.*
28. a) Write an exponential expression for the area of a rectangle with a length of [pic]meters and a width of [pic]meters. b) Evaluate the expression to find the area of the rectangle.*
Homework
29. 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. [pic]= 11?
i. 37 ÷ 39 = 3?
j. [pic] = 2?
k. [pic]= 132
l. 5? ÷ 56 = 53
30. A rectangle has a length of [pic]mm and a width of [pic]mm. Write an expression for the area of the rectangle as a power of 4.*
31. Express the volume of a cube with a side length of [pic]inches as a power of 2.*
32. a) Write an exponential expression for the area of a rectangle with a length of [pic] meters and a width of [pic]meters. b) Evaluate the expression to find the area of the rectangle.*
Roots & Radicals Review
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) [pic]
a. 14
b. 12
c. 72
d. 21
7) [pic]
a. 10
b. 6
c. 0.6
d. 18
8) [pic]
a. -7
b. 0.7
c. 0.07
d. -0.07
Circle whether the given number is rational or irrational
9) π rational irrational
10) 0.875 rational irrational
11) )[pic] rational irrational
12) Between what two integers does the following square root fall?
[pic]
a. 4 & 5
b. 6 & 7
c. 7 & 8
d. 5 & 6
13) (47)(43) = 4?
a. 10
b. 24
c. 4
d. 5
14) Approximate [pic] [pic] ________
15) Find the missing value 114 ÷ 116 = 11? ____________________
16) [pic]________________
17) (67)(6-2) = 6?
18) A rectangle has a length of [pic]cm and a width of [pic]cm. What is the area of the rectangle written as a power of 4? ___________________
19) Draw and label a number line from 0-10. Place the following numbers on the number line:* [pic],[pic],[pic]
20) Write [pic] as a fraction in simplest form.*
21) Write [pic] as a fraction in simplest form.*
22) 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.
Answer Key
1.
a. 3 units
b.
c. 3
d. Area of a Square = Side2 and 9 = 32
2.
|Side Length of a |Area of the square|
|Square |(units2) |
|(units) | |
|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.
4.
a. 5
b. 8
c. 9
d. 7
e. 4
5.
a. 6 units
b.
c. 6
d. Area = Side2 and 36 = 62
6.
|Side Length of a |Area of the square|
|square |(units2) |
|(units) | |
|14 |196 |
|15 |225 |
|16 |256 |
|17 |289 |
|18 |324 |
|19 |361 |
|20 |400 |
7.
a. 17
b. 20
c. 14
d. 19
e. 12
8.
|Side Length of a |Area of the square|
|square |(units2) |
|(units) | |
|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. (or 102 times greater).
10.
a. 53
b. 89
c. 22
d. 80
e. 45
f. 15
g. 29
h. 97
i. 31
j. 66
11.
a. 71
b. 36
c. 92
d. 55
e. 61
f. 83
g. 24
h. 49
i. 50
j. 17
12.
a. 5
b. 8
c. -9
d. No real solution
e. 7
f. [pic]
g. No real solution
h. ½
i. [pic]
j. – ½
k. 0.8
l. 0.09
m. -0.5
n. 0.04
o. No real solution
13.
a. 17
b. -20
c. 8
d. 19
e. No real solution
f. 1/3
g. ½
h. No real solution
i. -3/4
j. 1/10
k. No real solution
l. -0.14
m. 0.7
n. 0.19
o. 0.5
14.
a.
[pic]
b.
[pic]
c.
[pic]
d.
[pic]
e.
[pic]
15.
16.
a. 2.45
b. 8.37
c. 7.42
d. 3.74
e. 10.29
f. 6.4
g. 8.94
h. 8.06
i. 2.83
j. 15.26
17.
a. 7
b. 6
c. 8
d. 3
e. 9
18. 55
19. 26
20. 15
21.
a.
[pic]
b.
[pic]
c.
[pic]
d.
[pic]
e.
[pic]
22.
23.
a. 8.83
b. 2.65
c. 7.94
d. 5.39
e. 6.48
f. 11.75
g. 17.32
h. 12.17
i. 4.58
j. 7.21
24.
a. 4
b. 6
c. 4
d. 6
e. 7
25. 27
26. 37
27. 48
28.
a. Rational
b. Irrational
c. Irrational
d. Rational
e. Irrational
f. Rational
g. Irrational
h. Rational
i. Rational
j. Rational
29. c
30. a,b
31.
a. Rational
b. Irrational
c. Rational
d. Rational
e. Rational
f. Rational
g. Irrational
h. Irrational
i. Rational
j. Rational
32. a,c,d
33. a
34.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. 7
35.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
36.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. 4
37.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
f. [pic]
g. [pic]
h. [pic]
i. [pic]
38.
a. 7
b. 10
c. 3
d. 6
e. 8
f. 2
g. 2
h. 3
i. -3
j. -2
k. 5
l. 7
39. [pic]mm2
40. [pic]in3
41. a. [pic] x [pic] b. [pic]m2
42.
a. 9
b. 7
c. 2
d. 3
e. 8
f. 3
g. 4
h. 4
i. -2
j. -4
k. 4
l. 9
43. [pic]mm2
44. [pic]in3
45. a. [pic] x [pic] b. [pic]in2
Review answers
1. Yes
2. No
3. Yes
4. Yes
5. No
6. B
7. C
8. D
9. Irrational
10. rational
11. irrational
12. b
13. a
14. ~7
15. -2
16. 4
17. 5
18. [pic]cm2
19. [pic] =5.29
[pic] =5.48
[pic] =9.1
20. [pic]
21. [pic]
22. Expressions will vary
* From Engage NY
* From Engage NY
* From Engage NY
* From Engage NY
* From Engage NY
* From Engage NY
* From Engage NY
-----------------------
3 units
3 units
6 units
6 units
................
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