Find the square root of the following numbers by the prime factorisation method ( ix ) 8100
The Correct Answer and Explanation is:
To find the square root of 8100 using the prime factorisation method, follow these steps:
✅ Step-by-step Prime Factorisation of 8100:
Start by dividing 8100 by the smallest prime numbers:
- 8100÷2=40508100 \div 2 = 40508100÷2=4050
- 4050÷2=20254050 \div 2 = 20254050÷2=2025
- 2025÷3=6752025 \div 3 = 6752025÷3=675
- 675÷3=225675 \div 3 = 225675÷3=225
- 225÷3=75225 \div 3 = 75225÷3=75
- 75÷3=2575 \div 3 = 2575÷3=25
- 25÷5=525 \div 5 = 525÷5=5
- 5÷5=15 \div 5 = 15÷5=1
Now write the prime factorisation: 8100=22×34×528100 = 2^2 \times 3^4 \times 5^28100=22×34×52
✅ Finding the Square Root:
To find the square root, pair up the prime factors: 8100=22×34×52=2×32×5=2×9×5=90\sqrt{8100} = \sqrt{2^2 \times 3^4 \times 5^2} = 2 \times 3^2 \times 5 = 2 \times 9 \times 5 = 908100=22×34×52=2×32×5=2×9×5=90
🎯 Final Answer:
8100=90\sqrt{8100} = \boxed{90}8100=90
📘 Explanation
The prime factorisation method is a systematic approach to finding the square root of a number by expressing it as a product of prime numbers. The idea is to break the number down into its basic building blocks — its prime factors — and then use the rules of square roots to simplify.
In the case of 8100, we start by dividing it by the smallest prime number, which is 2, and continue dividing by prime numbers like 3 and 5 until we reach 1. This gives us: 8100=22×34×528100 = 2^2 \times 3^4 \times 5^28100=22×34×52
Next, we take the square root. Since we are finding the square root, we look for pairs of prime factors. Each pair of the same prime number contributes one of that prime to the square root.
For instance, 222^222 becomes 222, 343^434 becomes 32=93^2 = 932=9, and 525^252 becomes 555. Multiply these results: 2×9×5=902 \times 9 \times 5 = 902×9×5=90
Therefore, the square root of 8100 is 90.
This method is particularly useful for perfect squares, as the factorisation ensures all primes occur in even powers. If a prime factor doesn’t form a complete pair, then the number is not a perfect square. In this case, since all exponents were even, 8100 is a perfect square, and its root is an integer: 90.
