take the square root of problem square root 64^6 please show all work
262144
√x = x^1/2
√(64^6) = 64^3 = 262144
To calculate the square root of a number, you can follow these steps:
Step 1: Express the number as a product of its prime factors.
- First, let's find the prime factorization of 64:
- Divide 64 by the smallest prime number, which is 2. 64 ÷ 2 = 32.
- Divide 32 by 2 again: 32 ÷ 2 = 16.
- Divide 16 by 2 again: 16 ÷ 2 = 8.
- Divide 8 by 2 again: 8 ÷ 2 = 4.
- Divide 4 by 2 again: 4 ÷ 2 = 2.
- Divide 2 by 2 one last time: 2 ÷ 2 = 1.
- Therefore, the prime factorization of 64 is 2 × 2 × 2 × 2 × 2 × 2 = 2^6.
Step 2: Group the prime factors in pairs.
- In this case, since 6 is an even number, we can group the factors into pairs of 2's.
Step 3: Take one factor from each pair.
- By taking one factor from each pair, we are extracting the square root of the number.
- In this case, we have three pairs of 2's, so we take one 2 from each pair.
- This leaves us with an answer of 2 × 2 × 2 = 8.
Therefore, the square root of 64^6 is 8.