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:
- Divide by 2: 8100÷2=40504050÷2=20258100 \div 2 = 4050 \\ 4050 \div 2 = 20258100÷2=40504050÷2=2025 So, two 2s: 2×22 \times 22×2
- Divide by 3: 2025÷3=675675÷3=225225÷3=7575÷3=252025 \div 3 = 675 \\ 675 \div 3 = 225 \\ 225 \div 3 = 75 \\ 75 \div 3 = 252025÷3=675675÷3=225225÷3=7575÷3=25 So, four 3s: 3×3×3×33 \times 3 \times 3 \times 33×3×3×3
- Divide by 5: 25÷5=55÷5=125 \div 5 = 5 \\ 5 \div 5 = 125÷5=55÷5=1 So, two 5s: 5×55 \times 55×5
✨ Prime Factorisation of 8100:
8100=22×34×528100 = 2^2 \times 3^4 \times 5^28100=22×34×52
To find the square root, take half the powers of each prime factor:8100=22×34×52=21×32×51=2×9×5=90\sqrt{8100} = \sqrt{2^2 \times 3^4 \times 5^2} = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = \boxed{90}8100=22×34×52=21×32×51=2×9×5=90
📝 Explanation
The prime factorisation method is a systematic approach to finding the square root of a perfect square by breaking it down into its prime number components. Prime numbers are those that have only two distinct divisors: 1 and the number itself. The commonly used prime numbers include 2, 3, 5, 7, 11, etc.
We start by dividing 8100 by the smallest prime number, 2. We continue dividing until it is no longer divisible by 2. Then we proceed with the next smallest prime number (3), and so on, until the result becomes 1. By writing all these prime factors together, we get the prime factorisation of 8100:8100=22×34×528100 = 2^2 \times 3^4 \times 5^28100=22×34×52
To find the square root using this method, we apply the rule that:
a2=a\sqrt{a^2} = aa2=a
So, for each prime factor, we divide the exponent by 2. The square root of 810081008100 becomes:22×34×52=21×32×51=2×9×5=90\sqrt{2^2 \times 3^4 \times 5^2} = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 9022×34×52=21×32×51=2×9×5=90
This shows that the square root of 8100 is 90. This method ensures complete accuracy, especially useful when calculating square roots of large perfect squares without a calculator.
