square root of 512 simplified

Steps to Solve

1. Find the prime factors of 512

First, we need to factor 512 into its prime constituents. We can do this by continuously dividing by the smallest prime number, 2.

$$ 512 \div 2 = 256 $$
$$ 256 \div 2 = 128 $$
$$ 128 \div 2 = 64 $$
$$ 64 \div 2 = 32 $$
$$ 32 \div 2 = 16 $$
$$ 16 \div 2 = 8 $$
$$ 8 \div 2 = 4 $$
$$ 4 \div 2 = 2 $$
$$ 2 \div 2 = 1 $$

So, we find that the prime factors of 512 are (2^{9}), because we divided by 2 a total of 9 times.

2. Express 512 in terms of its prime factors

We can express 512 in terms of its prime factorization as:

$$ 512 = 2^9 $$

3. Simplify the square root

Next, we need to simplify the square root of (512) using its prime factors:

$$ \sqrt{512} = \sqrt{2^9} $$

Now, we can use the property of square roots, which states that ( \sqrt{a^b} = a^{b/2} ):

$$ \sqrt{2^9} = 2^{9/2} = 2^{4.5} $$

This means we can express it as (2^4 \times 2^{0.5}):

$$ 2^{4.5} = 2^4 \sqrt{2} = 16\sqrt{2} $$

4. Final result

Thus, the simplified form of the square root of 512 is:

$$ \sqrt{512} = 16\sqrt{2} $$