Find the square root of 400 by Prime Factorization Method.

To find the square root of 400 using the prime factorization method, we first need to break down the number 400 into its prime factors. This method is quite effective because it allows us to see the building blocks of the number, making it easier to find the square root.

Step 1: Prime Factorization of 400

We start by dividing 400 by the smallest prime number, which is 2. We continue dividing by 2 until we can no longer do so:

• 400 ÷ 2 = 200
• 200 ÷ 2 = 100
• 100 ÷ 2 = 50
• 50 ÷ 2 = 25

Now, 25 cannot be divided by 2, so we move to the next prime number, which is 3. Since 25 is not divisible by 3, we try the next prime number, 5:

• 25 ÷ 5 = 5
• 5 ÷ 5 = 1

Now we have reached 1, and we can summarize the prime factorization of 400:

Prime Factorization Result

The prime factorization of 400 is:

400 = 2^4 × 5^2

Step 2: Finding the Square Root

To find the square root, we take each prime factor and divide the exponent by 2:

• For 2^4: The square root is 2^(4/2) = 2^2 = 4
• For 5^2: The square root is 5^(2/2) = 5^1 = 5

Now, we multiply these results together to find the square root of 400:

Calculating the Final Result

Square root of 400 = 4 × 5 = 20

Verification

To ensure our answer is correct, we can square 20:

20 × 20 = 400

This confirms that our calculation is accurate. Therefore, the square root of 400, using the prime factorization method, is 20.