Name:__________________________________
th
8 Grade Math
5.2 Homework: Squares, Cubes and Irrational Numbers
Period: ____
Squares and Square Roots
When you multiply a number by itself, you square the number.
32 = 3 ∙ 3 = 9
To “undo” this, take the square root of the number. (The square root symbol,√
radical.)
√9 = √32 = 3
, is called a
1-18 The squares of whole numbers are called perfect squares. Write the first 10 perfect squares.
MEMORIZE THEM!! Then write the square roots of these numbers.
12 = 1 ∙ 1 = 1
62 = ____________
√1 = √12 = 1
√36 = ____________
22 = ____________
72 = ____________
√4 =____________
√49 = ____________
32 = ____________
82 = ____________
√9 = ____________
√64 = ____________
42 = ____________
92 = ____________
√16 = ___________
√81 = ____________
√25 = ___________
√144 = ___________
52 = ____________
102 = ___________
Area of Squares and Side Lengths
The area of a square is the space inside the square. Area is measured in square units.
The area of a square can be found by squaring its side. (𝐴 = 𝑠 2 )
The length of the side of a square can be found by taking the square root of its Area. (𝑠 = √𝐴)
𝐴 = 32 = 9 cm2
𝑠 = √9 = 3 cm
3 cm
3 cm
Use a square root symbol to write the side length of the square. Then find the side
length. Round to the nearest tenth.
19. 𝐴 = 625 in2
20. 𝐴 = 33.64 ft2
22. Solve. Write your answer in the most accurate form.
a. 𝑥 2 = 144
b. 𝑥 2 = 3
21. 𝐴 = 342.25 cm2
c. 𝑥 2 = 11
Cube Roots and Cubes
When you multiply a number by itself three times, you cube the number.
53 = 5 ∙ 5 ∙ 5 = 125
To “undo” this, take the cube root of the number.
3
3
√125 = √53 = 5
23-30 The cubes of whole numbers are called perfect cubes. Write the first 5 perfect
cubes. MEMORIZE THEM!! Then write the cube roots of these numbers.
13 = 1 ∙ 1 ∙ 1 = 1
3
23 = ________________________
3
33 = ________________________
3
43 = ________________________
3
53 = ________________________
3
3
√1 = √13 = 1
√8 = _________________________
√27 = ________________________
√64 = ________________________
√125 = _______________________
Volume of Cubes and Edge Lengths
The Volume of a cube is the 3-dimensional space inside the cube. Volume is measured in cubic
units. The volume of a cube can be found by cubing its edge. (𝑉 = 𝑒 3 )
3
The length of the edge of a cube can be found by taking the cube root of its volume. (𝑒 = √𝑉 )
3 cm
𝑉 = 32 = 27 cm3
3
𝑒 = √27 = 3 cm
3 cm
3 cm
Use a cube root symbol to write the edge length of the cube. Then find the edge length.
Round to the nearest tenth
31. 𝑉 = 512 in3
33. Solve. Write your answer in the most accurate form.
a. 𝑥 3 = 729
b. 𝑥 3 = 7
32. 𝑉 = 5832 cm3
c. 𝑥 3 = 216
Use your calculator to determine if the following numbers are PERFECT SQUARES (yes/no)
34. 100 _______
35. 144 _______
36. 150 _______
37. 200 ________
38. 169 _______
39. 300 _______
40. 225 _______
41. 625 ________
Place the following numbers on the number line (make sure to label the point).
42. √8
43. √36
44. −√4
45. √27
46. √20
47. √30
48. √15
49. −√10
3
3
3
Approximate Irrational Square Roots (determine which 2 integers the square root is in
between).
50. √5 is in between _____ & ______
51. √10 is in between _____ & ______
52. √2 is in between _____ & ______
53. √12 is in between _____ & ______
Approximate Irrational Cube Roots (determine which 2 integers the cube root is in between).
8
27
64
54. ∛45 is in between _____ & ______
55. ∛5 is in between _____ & ______
56. ∛18 is in between _____ & ______
57. ∛10 is in between _____ & ______
58. The following irrational numbers are between what two integers? No calculators. Show your
reasoning by showing your work. It might be helpful to make list of the perfect squares/cubes
and their roots.
Review
59. Convert the following terminating decimals to fractions. Always reduce to lowest terms.
60. Convert the following repeating decimals to fractions. Always reduce to lowest terms.
5.2 KEY
1. 4
3. 16
5. 36
7. 64
9. 100
11. 3
13. 5
15. 7
17. 9
19. 25 in.
21. 18.5 cm
23. 8
25. 64
27. 2
29. 4
31. 8 in.
33. a. 9
35. yes
37. no
39. no
41. yes
43. 6
45. 3
47. 3.107
49. -3.162
51. 3 & 4
53. 3 & 4
55. 1 & 2
57. 2 & 3
59. a. 233/1000