About Square Root and Cube Root
A square root asks which number multiplied by itself gives the given number, and a cube root asks which number used three times as a factor does so. Most questions are settled by knowing the squares up to 30 and the cubes up to 15.
What you need to understand
- The square root of a number is the value which multiplied by itself gives that number.
- The cube root is the value which multiplied by itself twice gives that number.
- A perfect square never ends in 2, 3, 7 or 8, which is a quick way to reject a number as a square.
- A perfect cube can end in any digit, and the last digit of the cube fixes the last digit of the root.
- The square root of a product is the product of the square roots, so two awkward roots often combine into a simple one.
- Squaring a number is not the same as doubling it, and halving a number is not the same as taking its root.
Formulas to remember
How to work through these questions
- Learn the squares up to 30 and the cubes up to 15, because most questions are drawn from that range.
- Use the last digit to narrow the answer down. A square ending in 9 has a root ending in 3 or in 7.
- For a product of two roots, multiply the numbers under the roots first and then take the root of the result.
- Estimate between the two nearest known squares to check the size of an answer.
- Multiply the root back out as a check, which takes only a moment.
Mistakes that cost marks
- Halving the number instead of taking its square root.
- Doubling a square root instead of squaring it.
- Taking the square when the question asked for the cube root.
- Assuming two awkward square roots cannot combine into a whole number.
- Mixing up entries in the squares table, which is the main source of careless error here.
Worked example
Find the value of the square root of 8 multiplied by the square root of 18.
- The square root of a product is the product of the square roots, so multiply the numbers under the roots first.
- 8 x 18 = 144.
- The square root of 144 is 12, because 12 x 12 = 144.
- Check: the square root of 8 is about 2.83 and the square root of 18 is about 4.24, and 2.83 x 4.24 is about 12.
Answer: 12
Practice questions with answers
A few Square Root and Cube Root questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Find the value of $\sqrt{3} \times \sqrt{27}$.
Answer: Option D — with explanation
The square root of a product is the product of the square roots, so multiply first.
$\sqrt{3} \times \sqrt{27} = \sqrt{3 \times 27} = \sqrt{81} = 9$.
Common mistakes
- multiplied the numbers and forgot the root, or added them instead: 30
- multiplied the numbers and forgot the root, or added them instead: 40.5
- multiplied the numbers and forgot the root, or added them instead: 81
Question 2
Find the value of $\sqrt{5} \times \sqrt{45}$.
Answer: Option D — with explanation
The square root of a product is the product of the square roots, so multiply first.
$\sqrt{5} \times \sqrt{45} = \sqrt{5 \times 45} = \sqrt{225} = 15$.
Common mistakes
- multiplied the numbers and forgot the root, or added them instead: 225
- multiplied the numbers and forgot the root, or added them instead: 50
- multiplied the numbers and forgot the root, or added them instead: 112.5
Question 3
Find the value of $\sqrt{20} \times \sqrt{45}$.
Answer: Option C — with explanation
The square root of a product is the product of the square roots, so multiply first.
$\sqrt{20} \times \sqrt{45} = \sqrt{20 \times 45} = \sqrt{900} = 30$.
Common mistakes
- multiplied the numbers and forgot the root, or added them instead: 450
- multiplied the numbers and forgot the root, or added them instead: 900
- multiplied the numbers and forgot the root, or added them instead: 65
Question 4
Work out $\sqrt{10} \times \sqrt{40}$.
Answer: Option B — with explanation
The square root of a product is the product of the square roots, so multiply first.
$\sqrt{10} \times \sqrt{40} = \sqrt{10 \times 40} = \sqrt{400} = 20$.
Common mistakes
- multiplied the numbers and forgot the root, or added them instead: 400
- multiplied the numbers and forgot the root, or added them instead: 200
- multiplied the numbers and forgot the root, or added them instead: 50
Question 5
What is the value of $\sqrt{8} \times \sqrt{18}$?
Answer: Option B — with explanation
The square root of a product is the product of the square roots, so multiply first.
$\sqrt{8} \times \sqrt{18} = \sqrt{8 \times 18} = \sqrt{144} = 12$.
Common mistakes
- multiplied the numbers and forgot the root, or added them instead: 72
- multiplied the numbers and forgot the root, or added them instead: 26
- multiplied the numbers and forgot the root, or added them instead: 144
Frequently asked questions
How do I take the square root of a product of two roots?
Multiply the numbers under the roots and take the root of the result. Two awkward roots often multiply into a perfect square, as 8 and 18 do.
Which numbers cannot be perfect squares?
A perfect square never ends in 2, 3, 7 or 8. If a number ends in one of those digits, it is not a perfect square.
How do I find a cube root quickly?
Use the last digit. A cube ending in 3 has a root ending in 7, and a cube ending in 7 has a root ending in 3. Then estimate the size from the leading digits.
What is the difference between squaring and doubling?
Squaring multiplies a number by itself, while doubling adds it to itself. Three squared is nine, but three doubled is six.
