simplify the square root of 52

Steps to Solve

1. Factor 52 into prime factors

To simplify the square root of 52, we start by finding its prime factorization. The number 52 can be factored as:

$$ 52 = 2 \times 26 $$

Then, we can factor 26 further:

$$ 26 = 2 \times 13 $$

So, the complete factorization of 52 is:

$$ 52 = 2^2 \times 13 $$

2. Identify the perfect square

Next, we identify the perfect square within the factorization. In this case, $2^2$ is a perfect square.

3. Apply the square root property

Using the property of square roots, we can split the square root of a product:

$$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $$

So we apply this to our factors:

$$ \sqrt{52} = \sqrt{2^2 \times 13} = \sqrt{2^2} \times \sqrt{13} $$

4. Simplify the square root

Now we can simplify the square root of $2^2$:

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

So we have:

$$ \sqrt{52} = 2 \times \sqrt{13} $$

5. Final Expression

Thus, the simplified form of $\sqrt{52}$ is:

$$ \sqrt{52} = 2\sqrt{13} $$