Square roots by prime factorisation: Class 8 practice
A perfect square has every prime factor in pairs. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Prime factorisation
The idea behind square roots by prime factorisation
A perfect square has every prime factor in pairs. Take one factor from each pair to get the square root.
Step-by-step method
- Prime-factorise the number.
- Group equal factors in pairs.
- Multiply one from each pair.
- If a factor is unpaired, multiply (or divide) by it to make a perfect square.
Worked example 1 (Basic)
Find √324 by prime factorisation.
- 324 = 2 × 2 × 3 × 3 × 3 × 3
- One from each pair: 2 × 3 × 3 = 18
- Answer: 18
Worked example 2 (Standard)
Find √1024 by prime factorisation.
- 1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
- One from each pair: 2 × 2 × 2 × 2 × 2 = 32
- Answer: 32
Worked example 3 (Challenge)
Find the smallest number by which 6480 must be multiplied to make it a perfect square, and the square root of the result.
- 6480 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 5
- 5 is unpaired → multiply by 5
- √32400 = 180
- Answer: 5; √32400 = 180
Common mistakes and misconceptions
- Taking both factors from a pair
- Assuming numbers ending in 2, 3, 7, 8 can be perfect squares
Practice questions
- (Basic) Find √256 by prime factorisation.
- (Basic) Find √361 by prime factorisation.
- (Standard) Find √784 by prime factorisation.
- (Standard) Find √400 by prime factorisation.
- (Challenge) Find the smallest number by which 7220 must be multiplied to make it a perfect square, and the square root of the result.
- (Challenge) Find the smallest number by which 1575 must be multiplied to make it a perfect square, and the square root of the result.
Answer key with full working
- Q1: 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 → One from each pair: 2 × 2 × 2 × 2 = 16 → Answer: 16
- Q2: 361 = 19 × 19 → One from each pair: 19 = 19 → Answer: 19
- Q3: 784 = 2 × 2 × 2 × 2 × 7 × 7 → One from each pair: 2 × 2 × 7 = 28 → Answer: 28
- Q4: 400 = 2 × 2 × 2 × 2 × 5 × 5 → One from each pair: 2 × 2 × 5 = 20 → Answer: 20
- Q5: 7220 = 2 × 2 × 5 × 19 × 19 → 5 is unpaired → multiply by 5 → √36100 = 190 → Answer: 5; √36100 = 190
- Q6: 1575 = 3 × 3 × 5 × 5 × 7 → 7 is unpaired → multiply by 7 → √11025 = 105 → Answer: 7; √11025 = 105
Syllabus note
This skill is listed under "Squares and Square Roots" for Class 8 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is square roots by prime factorisation in?
"Squares and Square Roots", Class 8 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.