Squares & Square Roots — Challenge
Squaring and taking a square root undo each other. A root asks: what number, times itself, gives this?
What these problems 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
What is √81?
- 8
- 9
- 18
- 40.5
- 81
Worked answer
9
A square root asks which number multiplied by itself gives 81.
9· 9 = 81, so √81 = 9. Halving instead would give 40.5.
Problem 2
Write √50 in simplest radical form.
- 5√2
- 2√5
- 10√5
- 25√2
- √10
Worked answer
5√2
Look for the largest perfect square that divides 50. That is 25, since 50 = 25· 2.
√50 = √25·√2 = 5√2.
As a check, 5√2≈ 7.07 and 7.072≈ 50.
Problem 3
What is √1449?
- 4
- 12
- 16
- 48
- 43
Worked answer
4
The square root of a fraction is the root of the top over the root of the bottom.
√144√9 = 123 = 4.
Equivalently, 1449 = 16 and √16 = 4.
Problem 4
What is √49+√16?
- 9
- 11
- 13
- 65
- √65
Worked answer
11
Take each root separately: √49 = 7 and √16 = 4.
7+4 = 11. Note that √49+16 = √65≈ 8.06, which is not the same thing: roots cannot be added inside the sign.
Problem 5
Between which two consecutive whole numbers does √60 lie?
- 6 and 7
- 7 and 8
- 8 and 9
- 9 and 10
- 30 and 31
Worked answer
7 and 8
Find the perfect squares either side of 60: 49 and 64.
Since 49 < 60 < 64, taking roots gives 7 < √60 < 8.
So √60 lies between 7 and 8, closer to 8 because 60 is closer to 64.
Problem 6
Write √72 in simplest radical form.
- 6√2
- 2√6
- 3√6
- 8√3
- 36√2
Worked answer
6√2
The largest perfect square dividing 72 is 36, because 72 = 36· 2.
√72 = √36·√2 = 6√2.
Using 4 instead gives 2√18, which is correct but not simplest, since 18 still contains the square 9.
Problem 7
What is √121?
- 10
- 11
- 12
- 60.5
- 121
Worked answer
11
Ask which number times itself makes 121.
11· 11 = 121, so √121 = 11.
Problem 8
What is (√5)2?
- √5
- 5
- 10
- 25
- √10
Worked answer
5
Squaring undoes a square root, because the root is defined as the number whose square is 5.
So (√5)2 = 5 exactly, with no rounding.
Problem 9
What is √8·√2?
- 4
- √10
- 2√2
- 8
- 16
Worked answer
4
Roots multiply under one sign: √8·√2 = √16.
√16 = 4. Adding the numbers inside instead would give √10, which is about 3.16.
Problem 10
What is √36?
- 4
- 6
- 9
- 18
- 36
Worked answer
6
Ask which number times itself makes 36.
6· 6 = 36, so √36 = 6.