Square Root by Prime Factorisation
Finding the square root of a number means finding a value that, when multiplied by itself, gives the original number. For example, the square root of 144 is 12 because 12 x 12 = 144.
The prime factorisation method is one of the most systematic ways to find the square root of a perfect square. In this method, you break the number into its prime factors, pair them up, and take one factor from each pair. The product of these factors gives the square root.
This method works only for perfect squares — numbers that are exact squares of integers. It is part of the Class 8 chapter "Squares and Square Roots" and is frequently tested in NCERT-based examinations.
What is Square Root by Prime Factorisation?
Definition: The prime factorisation method of finding square roots involves expressing the number as a product of prime factors, grouping them into pairs of identical factors, and multiplying one factor from each pair.
Key terms:
- Prime factorisation: Expressing a number as a product of prime numbers (e.g., 36 = 2 x 2 x 3 x 3).
- Perfect square: A number whose prime factorisation has all primes in pairs (e.g., 36 = 2² x 3²).
- Square root: The number obtained by taking one factor from each pair (e.g., sqrt(36) = 2 x 3 = 6).
Important: If any prime factor appears an odd number of times, the number is NOT a perfect square, and this method cannot give an exact integer square root.
Square Root by Prime Factorisation Formula
Method:
Step 1: Find prime factorisation
Step 2: Make pairs of identical prime factors
Step 3: Take one factor from each pair
Step 4: Multiply them to get the square root
Example:
- 324 = 2 x 2 x 3 x 3 x 3 x 3
- Pairs: (2 x 2) and (3 x 3) and (3 x 3)
- Take one from each pair: 2 x 3 x 3 = 18
- sqrt(324) = 18
Verification: 18 x 18 = 324. Correct.
Derivation and Proof
Why does this method work?
If a number N is a perfect square, then N = k², where k is some integer.
Express N in prime factorisation:
- N = p₁^a₁ x p₂^a₂ x p₃^a₃ x ...
Since N = k², each exponent a₁, a₂, a₃, ... must be even. (If any exponent were odd, N would not be a perfect square.)
Then:
- k = p₁^(a₁/2) x p₂^(a₂/2) x p₃^(a₃/2) x ...
This is exactly what we do when we "take one from each pair" — we halve each exponent.
Example with exponents:
- 7056 = 2⁴ x 3² x 7²
- sqrt(7056) = 2² x 3¹ x 7¹ = 4 x 3 x 7 = 84
- Verification: 84² = 7056. Correct.
Types and Properties
The prime factorisation method can be used in several types of problems:
1. Finding square root of a perfect square:
- Factorise, pair up, take one from each pair, multiply.
2. Checking if a number is a perfect square:
- Factorise the number. If all prime factors appear an even number of times, it is a perfect square.
3. Finding the smallest number to multiply/divide to make a perfect square:
- Factorise the number. Identify the prime(s) that appear an odd number of times.
- To make it a perfect square by multiplication, multiply by those primes.
- To make it a perfect square by division, divide by those primes.
4. Square root of a fraction:
- Find the square root of the numerator and denominator separately.
- sqrt(a/b) = sqrt(a) / sqrt(b)
5. Square root of a product:
- sqrt(a x b) = sqrt(a) x sqrt(b)
- Factorise the entire product and apply the method.
Solved Examples
Example 1: Example 1: Square root of 576
Problem: Find the square root of 576 by prime factorisation.
Solution:
Prime factorisation:
- 576 / 2 = 288
- 288 / 2 = 144
- 144 / 2 = 72
- 72 / 2 = 36
- 36 / 2 = 18
- 18 / 2 = 9
- 9 / 3 = 3
- 3 / 3 = 1
576 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 = 2⁶ x 3²
Pairing: (2 x 2)(2 x 2)(2 x 2)(3 x 3)
Take one from each pair: 2 x 2 x 2 x 3 = 24
Answer: sqrt(576) = 24.
Example 2: Example 2: Square root of 1296
Problem: Find sqrt(1296) by prime factorisation.
Solution:
Prime factorisation:
- 1296 = 2 x 648 = 2 x 2 x 324 = 2 x 2 x 2 x 162 = 2 x 2 x 2 x 2 x 81
- 81 = 3 x 3 x 3 x 3
1296 = 2⁴ x 3⁴
Take one from each pair: 2² x 3² = 4 x 9 = 36
Answer: sqrt(1296) = 36.
Example 3: Example 3: Square root of 3136
Problem: Find sqrt(3136) by prime factorisation.
Solution:
Prime factorisation:
- 3136 / 2 = 1568
- 1568 / 2 = 784
- 784 / 2 = 392
- 392 / 2 = 196
- 196 / 2 = 98
- 98 / 2 = 49
- 49 / 7 = 7
- 7 / 7 = 1
3136 = 2⁶ x 7²
Take one from each pair: 2³ x 7 = 8 x 7 = 56
Answer: sqrt(3136) = 56.
Example 4: Example 4: Is 2352 a perfect square?
Problem: Is 2352 a perfect square? If not, find the smallest number by which it should be multiplied to make it a perfect square.
Solution:
Prime factorisation:
- 2352 = 2 x 2 x 2 x 2 x 3 x 7 x 7
- 2352 = 2⁴ x 3¹ x 7²
The prime factor 3 appears only once (odd). So 2352 is NOT a perfect square.
To make it a perfect square, multiply by 3:
- 2352 x 3 = 7056 = 2⁴ x 3² x 7²
- sqrt(7056) = 2² x 3 x 7 = 84
Answer: 2352 is not a perfect square. Multiply by 3 to get 7056, whose square root is 84.
Example 5: Example 5: Smallest number to divide
Problem: Find the smallest number by which 1800 must be divided to make it a perfect square. Also find the square root of the resulting number.
Solution:
Prime factorisation:
- 1800 = 2 x 2 x 2 x 3 x 3 x 5 x 5
- 1800 = 2³ x 3² x 5²
The prime factor 2 appears 3 times (odd). Divide by 2:
- 1800 / 2 = 900 = 2² x 3² x 5²
- sqrt(900) = 2 x 3 x 5 = 30
Answer: Divide by 2 to get 900. sqrt(900) = 30.
Example 6: Example 6: Square root of a large number
Problem: Find the square root of 7056.
Solution:
Prime factorisation:
- 7056 / 2 = 3528
- 3528 / 2 = 1764
- 1764 / 2 = 882
- 882 / 2 = 441
- 441 / 3 = 147
- 147 / 3 = 49
- 49 / 7 = 7
- 7 / 7 = 1
7056 = 2⁴ x 3² x 7²
Take one from each pair: 2² x 3 x 7 = 4 x 3 x 7 = 84
Answer: sqrt(7056) = 84.
Example 7: Example 7: Square root of a fraction
Problem: Find the square root of 256/625.
Solution:
Numerator: 256 = 2⁸, so sqrt(256) = 2⁴ = 16
Denominator: 625 = 5⁴, so sqrt(625) = 5² = 25
sqrt(256/625) = 16/25
Answer: sqrt(256/625) = 16/25.
Example 8: Example 8: Product of two numbers
Problem: Find the square root of the product 441 x 196 without computing the product.
Solution:
sqrt(441 x 196) = sqrt(441) x sqrt(196)
Finding sqrt(441):
- 441 = 3² x 7²
- sqrt(441) = 3 x 7 = 21
Finding sqrt(196):
- 196 = 2² x 7²
- sqrt(196) = 2 x 7 = 14
sqrt(441 x 196) = 21 x 14 = 294
Answer: sqrt(441 x 196) = 294.
Example 9: Example 9: Square root of 11025
Problem: Find the square root of 11025.
Solution:
Prime factorisation:
- 11025 / 3 = 3675
- 3675 / 3 = 1225
- 1225 / 5 = 245
- 245 / 5 = 49
- 49 / 7 = 7
- 7 / 7 = 1
11025 = 3² x 5² x 7²
Take one from each pair: 3 x 5 x 7 = 105
Answer: sqrt(11025) = 105.
Example 10: Example 10: Finding the length of a square field
Problem: A square field has an area of 5625 m². Find the length of each side.
Solution:
Side of a square field = sqrt(Area) = sqrt(5625)
Prime factorisation:
- 5625 = 3 x 3 x 5 x 5 x 5 x 5 = 3² x 5⁴
Take one from each pair: 3 x 5² = 3 x 25 = 75
Answer: Each side of the field is 75 m.
Real-World Applications
The prime factorisation method for finding square roots is used in many situations:
- Geometry: Finding the side of a square when the area is given (side = sqrt(area)).
- Number Theory: Checking whether a number is a perfect square.
- Simplifying Surds: Simplifying expressions like sqrt(72) = sqrt(36 x 2) = 6sqrt(2) requires prime factorisation.
- HCF and LCM: Prime factorisation is the basis for finding HCF and LCM, and these connect to square root problems.
- Pythagoras Theorem: When the hypotenuse² gives a value like 1225, you need to find sqrt(1225) = 35.
- Statistics: Standard deviation calculations involve square roots of sums of squares.
- Real-life measurement: Calculating the side of a square plot, the radius from the area of a circle, or the edge of a cube from its volume.
Key Points to Remember
- This method works only for perfect squares.
- Steps: Prime factorise → Make pairs → Take one from each pair → Multiply.
- A number is a perfect square if and only if all prime factors appear an even number of times.
- If a prime factor appears an odd number of times, the number is NOT a perfect square.
- To make a non-perfect-square into a perfect square, multiply or divide by the unpaired prime factor(s).
- sqrt(a x b) = sqrt(a) x sqrt(b).
- sqrt(a/b) = sqrt(a) / sqrt(b).
- The prime factorisation method is best for numbers that are not too large.
- For very large numbers, the long division method is more practical.
- Always verify: square the answer and check if you get the original number.
Practice Problems
- Find the square root of 784 by prime factorisation.
- Find the square root of 4096 by prime factorisation.
- Is 1728 a perfect square? If not, find the smallest number to multiply to make it one.
- Find the smallest number by which 3675 must be divided to get a perfect square.
- Find sqrt(2025) by prime factorisation.
- Find the square root of 144/169.
- A square garden has area 1764 m². Find the length of each side.
- Find sqrt(12544) by prime factorisation.
Frequently Asked Questions
Q1. What is the prime factorisation method for finding square roots?
Express the number as a product of prime factors, group them into pairs of identical primes, take one prime from each pair, and multiply them together. The result is the square root.
Q2. Does this method work for all numbers?
No. This method gives an exact answer only for perfect squares — numbers where all prime factors appear an even number of times.
Q3. How do you check if a number is a perfect square using prime factorisation?
Find the prime factorisation. If every prime factor appears an even number of times, the number is a perfect square. If any prime factor appears an odd number of times, it is not.
Q4. What if a prime factor appears an odd number of times?
The number is not a perfect square. To make it one, multiply by that prime (to make its count even) or divide by it.
Q5. How do you find the square root of a fraction?
Find the square root of the numerator and denominator separately. sqrt(a/b) = sqrt(a) / sqrt(b). Both must be perfect squares for an exact answer.
Q6. What is the advantage of this method?
It is systematic, easy to understand, and also tells you whether the number is a perfect square. It also helps find the smallest multiplier/divisor to make a number a perfect square.
Q7. When should you use the long division method instead?
For very large numbers or for non-perfect squares (to find approximate square roots), the long division method is more practical.
Q8. Can this method find the square root of a decimal?
Not directly. Convert the decimal to a fraction first (e.g., 0.0225 = 225/10000), then apply the method to numerator and denominator separately.
Q9. How do you verify the answer?
Square the answer and check: if (answer)² = original number, the square root is correct.
Q10. What is the connection between prime factorisation and the exponent method?
If N = p₁^a₁ x p₂^a₂ x ..., then sqrt(N) = p₁^(a₁/2) x p₂^(a₂/2) x .... Taking one from each pair is the same as halving each exponent.
