Squares & Square Roots — Practice
Squaring and taking a square root undo each other. A root asks: what number, times itself, gives this?
What these questions cover
These questions use 4 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Area — Space inside, measured in square units.
- Perfect square — A number that is some whole number squared: 1, 4, 9, 16, 25, …
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Roots — The x values where y = 0.
Problem 1
Find √64.
Worked answer
8
8 × 8 = 64
Problem 2
Find √121.
Worked answer
11
11 × 11 = 121
Problem 3
Evaluate 92.
Worked answer
81
9 × 9 = 81
Problem 4
Find √0.25.
Worked answer
0.5
0.5 × 0.5 = 0.25
Problem 5
Is 50 a perfect square?
- Yes
- No
Worked answer
No
50 sits between 49 and 64, so no whole number squares to it.
Problem 6
√40 lies between which two consecutive whole numbers?
- 5 and 6
- 6 and 7
- 7 and 8
- 19 and 21
Worked answer
6 and 7
36 < 40 < 49
62 = 36 and 72 = 49
So 6 < √40 < 7
Problem 7
Simplify √72 into the form a√2. What is a?
Worked answer
6
72 = 36 × 2
√72 = √36 × √2 = 6√2
Problem 8
Solve x2 = 49.
- x = 7
- x = -7
- x = ± 7
- x = 24.5
Worked answer
x = ± 7
72 = 49 and (-7)2 = 49
An equation keeps both, so x = ± 7
Note √49 on its own means just 7.
Problem 9
Find √925.
Worked answer
3/5
√9√25 = 35
Problem 10
Compute √144 + √25.
Worked answer
17
12 + 5 = 17
Note this is NOT √169 = 13 — roots do not add that way.
Problem 11
Estimate √85 to the nearest whole number.
Worked answer
9
81 < 85 < 100, so between 9 and 10
85 is much closer to 81
√85 ≈ 9.2, so 9
Problem 12
A square has area 169 cm2. Find its side length.
Worked answer
13
√169 = 13 cm
Problem 13
Simplify √98 into the form a√2. What is a?
Worked answer
7
√98 = √49 × √2
= 7√2
Problem 14
Which is larger, √16 + 9 or √16 + √9?
- √16 + 9
- √16 + √9
- They are equal
Worked answer
√16 + √9
√16 + 9 = √25 = 5
√16 + √9 = 4 + 3 = 7
7 > 5, so √a+b ≠ √a + √b.
Problem 15
Solve x2 = 0.36. Give the positive solution.
Worked answer
0.6
x = ± 0.6
The positive solution is 0.6.
Problem 16
A square has area 200 cm2. Find its side to the nearest tenth of a cm.
Worked answer
14.1
142 = 196 and 14.22 = 201.6
14.12 = 198.81, closest to 200
Side ≈ 14.1 cm