Find the square roots of the numbers by expressing each number as the product of primes :
64

Answer

To find the square root of 64 by expressing it as the product of primes, we can follow these steps:

### Step 1: Prime Factorization of 64
We start by finding the prime factors of 64. We can do this by dividing 64 by the smallest prime number, which is 2.

- 64 ÷ 2 = 32
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1

So, we can express 64 as:
\[ 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \]
or in exponential form:
\[ 64 = 2^6 \]

### Step 2: Grouping the Prime Factors
Next, we group the prime factors into pairs since we are looking for the square root. Each pair of 2s will contribute one 2 to the square root.

From \( 2^6 \), we can form 3 pairs of 2s:
- Pair 1: \( 2 \times 2 \)
- Pair 2: \( 2 \times 2 \)
- Pair 3: \( 2 \times 2 \)

### Step 3: Taking the Square Root
Now, we take the square root of 64:
\[ \sqrt{64} = \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2)} \]

When we take the square root, each pair contributes one factor of 2:
\[ \sqrt{64} = 2 \times 2 \times 2 = 8 \]

### Final Answer
Thus, the square root of 64 is:
\[ \sqrt{64} = 8 \]

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