8100 Find the square root of this number by prime factorisation method.
The Correct Answer and Explanation is:
To find the square root of 8100 using the prime factorization method, follow these steps:
✅ Step 1: Prime Factorization of 8100
We start dividing 8100 by the smallest prime numbers:8100÷2=40504050÷2=20252025÷3=675675÷3=225225÷3=7575÷3=2525÷5=55÷5=1\begin{align*} 8100 \div 2 &= 4050 \\ 4050 \div 2 &= 2025 \\ 2025 \div 3 &= 675 \\ 675 \div 3 &= 225 \\ 225 \div 3 &= 75 \\ 75 \div 3 &= 25 \\ 25 \div 5 &= 5 \\ 5 \div 5 &= 1 \\ \end{align*}8100÷24050÷22025÷3675÷3225÷375÷325÷55÷5=4050=2025=675=225=75=25=5=1
So, the prime factorization of 8100 is:8100=22×34×528100 = 2^2 \times 3^4 \times 5^28100=22×34×52
✅ Step 2: Pair the Prime Factors
Since we are finding the square root, we look for pairs of prime factors:8100=22×34×52\sqrt{8100} = \sqrt{2^2 \times 3^4 \times 5^2}8100=22×34×52
Take one number from each pair:8100=2×32×5=2×9×5=90\sqrt{8100} = 2 \times 3^2 \times 5 = 2 \times 9 \times 5 = 908100=2×32×5=2×9×5=90
🎯 Answer: 8100=90\sqrt{8100} = \boxed{90}8100=90
🧠 Explanation
The prime factorization method is a systematic way to find the square root of a perfect square by breaking it down into its prime components. In this method, we continuously divide the number by its smallest prime factors until we reach 1. Each time we divide by a prime, we record it. Once we have completely factored the number, we express it as a product of its prime factors using exponents.
For 8100, we factorized it as:8100=22×34×528100 = 2^2 \times 3^4 \times 5^28100=22×34×52
Now, the square root of a number is a value that, when multiplied by itself, gives the original number. So when we take the square root of a number expressed in terms of prime factors, we take half of the power of each prime (since squaring doubles the powers, taking the square root halves them). So:8100=22×34×52=21×32×51\sqrt{8100} = \sqrt{2^2 \times 3^4 \times 5^2} = 2^{1} \times 3^{2} \times 5^{1}8100=22×34×52=21×32×51
Multiplying these gives:2×9×5=902 \times 9 \times 5 = 902×9×5=90
This method is particularly helpful for students as it builds a clear understanding of how square roots and exponents work, especially with large numbers. It’s also useful when calculators are not allowed in exams. In conclusion, the square root of 8100 by prime factorization is 90.
