find the square root of following number by long division method ,18225
find the square root of following number by long division method ,18225
The Correct Answer and Explanation is:
✅ Square Root of 18225 Using Long Division Method
Let’s find the square root of 18225 step by step using the long division method.
🔢 Step 1: Pair the digits
Start pairing the digits of 18225 from the right:
- 1 82 25 → we get three pairs: (1), (82), (25)
🔢 Step 2: Find the largest square ≤ first pair
First pair = 1
- √1 = 1 → 1² = 1
Write 1 as the first digit of the square root.
Quotient so far: 1
Now:
- Subtract: 1 – 1 = 0
- Bring down next pair: 82 → Now working on 82
🔢 Step 3: Double the quotient
Current quotient = 1 → Double = 2
We need a digit X such that:
(20 + X) × X ≤ 82
Try:
- X = 3 → (20 + 3) × 3 = 23 × 3 = 69 ✅
- X = 4 → 24 × 4 = 96 ❌ (too big)
So, X = 3
Write 3 next to 1 → Quotient = 13
Now:
- Subtract: 82 – 69 = 13
- Bring down next pair: 25 → Now working on 1325
🔢 Step 4: Double the quotient
Current quotient = 13 → Double = 26
We now find X such that:
(260 + X) × X ≤ 1325
Try:
- X = 5 → 265 × 5 = 1325 ✅
So, X = 5
Final quotient = 135
Now:
- Subtract: 1325 – 1325 = 0 → No remainder
✅ Final Answer:
√18225 = 135
📘 Explanation
The long division method is a systematic process used to find the square root of large numbers without using a calculator. It is especially helpful when perfect squares are involved, like 18225.
We begin by pairing digits of the number from right to left. For 18225, we group it as (1)(82)(25). These pairs help us break the number into manageable parts as we work through the root step by step.
We then find the largest square that fits into the first group (1). Since 1 is a perfect square, we start with 1. This becomes the first digit of our answer. After subtracting and bringing down the next pair (82), we proceed to the next stage.
We now double the root found so far (1 becomes 2), and find a digit (say X) that, when added to 20 (making 20 + X), and then multiplied by X, gives a product close to or equal to the current number (82). We find that 3 fits because 23 × 3 = 69. This 3 becomes the next digit in the square root.
We repeat this process for the next pair. Double 13 (which is our current root so far), giving 26. Now, we look for X such that (260 + X) × X is as close to 1325 as possible. X = 5 works perfectly here.
Once the entire number is exhausted and no remainder remains, the digits collected (135) form the square root. Hence, the square root of 18225 is 135.
This method is reliable and provides a clear, step-by-step approach for perfect square roots.
