By splitting into prime factors, find the square root of :
`194481`

Options

A. `321`
B. `421`
C. `441`
D. `299`

Answer

To find the square root of 194481 by splitting it into prime factors, we will follow these steps:

### Step 1: Prime Factorization
We need to find the prime factors of 194481. We will start dividing by the smallest prime numbers.

1. **Check divisibility by 7:**
- 194481 ÷ 7 = 27854.4286 (not divisible)

2. **Check divisibility by 3:**
- 194481 ÷ 3 = 64827 (divisible)

3. **Continue factoring 64827:**
- 64827 ÷ 3 = 21609
- 21609 ÷ 3 = 7203
- 7203 ÷ 3 = 2401

4. **Now factor 2401:**
- 2401 ÷ 7 = 343
- 343 ÷ 7 = 49
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1

### Step 2: Collecting the Prime Factors
From the above steps, we can collect the prime factors:
- 194481 = 3^4 × 7^4

### Step 3: Finding the Square Root
To find the square root, we take the square root of each prime factor:
- √(3^4) = 3^(4/2) = 3^2 = 9
- √(7^4) = 7^(4/2) = 7^2 = 49

### Step 4: Multiply the Results
Now, we multiply the results of the square roots:
- Square root of 194481 = 9 × 49 = 441

Thus, the square root of 194481 is **441**.

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