Square Roots and Cube Roots

Show your work for each problem.

1.
What is ?
1. a)4
2. b)18
3. c)9
4. d)6
Answer: 6
because . Note: 18 is half of 36 (division error), 9 is , and 4 is .
2.
What is ?
Answer: 7
because . Remember: 49 is a perfect square.
3.
What is ?
1. a)5.2
2. b)9
3. c)3
4. d)4
Answer: 3
because . Note: 9 is (division instead of cube root), 5.2 is (wrong root type).
4.
What is ?
Answer: 2
because . This is the first non-trivial perfect cube!
5.
Find
Answer: 10
1. We need to find a number that, multiplied by itself, equals 100. What is ? 100
2. Since , what is ? 10
6.
Which number is between and ?
1. a)7
2. b)6
3. c)8
4. d)9
Answer: 8
Since is between 60 and 70, and , the answer is 8. Also, and , so 8 is between them.
7.
A square has an area of 121 square meters. What is the length of each side?
Answer: 11
Side = meters. Check: ✓
8.
Find
Answer: 5
1. We need to find a number that, when cubed, equals 125. What is ? 64
2. 64 is too small. What is ? 125
3. Since , what is ? 5
9.
What is ?
1. a)14
2. b)12
3. c)72
4. d)11
Answer: 12
because . Note: 14 is close but , 72 is , and .
10.
A cube has a volume of 512 cubic centimeters. What is the edge length?
Answer: 8
1. Volume of a cube = edge. So edge = . We need . Is ? Calculate . 343
2. 343 is too small. What is ? 512
3. Since , what is the edge length in cm? 8
11.
What is ?
Answer: 6
because .
12.
Estimate to the nearest whole number.
Answer: 9
1. What is ? 64
2. What is ? 81
3. 75 is between 64 and 81. Is 75 closer to 64 or 81? 81
4. Since 75 is closer to 81, what is to the nearest whole number? 9
13.
What is ?
1. a)3 + 4
2. b)5
3. c)7
4. d)25
Answer: 5
. Common mistake: thinking it equals , but you cannot split roots over addition!
14.
What is ?
Answer: 10
because . This is one of the most important perfect cubes to memorize!
15.
A rectangular room has dimensions 5 meters by 12 meters. What is the length of the diagonal?
Answer: 13
1. By the Pythagorean theorem, diagonal = . What is ? 25
2. What is ? 144
3. What is ? 169
4. The diagonal = . What is ? 13
16.
What is ?
1. a)8
2. b)27
3. c)9
4. d)81
Answer: 9
because . Note: 27 is (wrong root), 81 is , and .