To simplify the expression [tex]\(\sqrt{252 g^4 h^8}\)[/tex], follow these steps:
1. Factor the radicand (the expression inside the square root):
First, factor the number [tex]\(252\)[/tex] into its prime factors:
[tex]\[
252 = 2^2 \times 3^2 \times 7
\][/tex]
So, we can rewrite the original expression as:
[tex]\[
\sqrt{252 g^4 h^8} = \sqrt{2^2 \times 3^2 \times 7 \times g^4 \times h^8}
\][/tex]
2. Simplify the square root:
The square root of a product is the product of the square roots of the factors. For the given expression, apply the square root to each factor separately:
[tex]\[
\sqrt{2^2 \times 3^2 \times 7 \times g^4 \times h^8} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{7} \times \sqrt{g^4} \times \sqrt{h^8}
\][/tex]
3. Simplify the square roots of the factors:
- [tex]\(\sqrt{2^2} = 2\)[/tex]
- [tex]\(\sqrt{3^2} = 3\)[/tex]
- [tex]\(\sqrt{7}\)[/tex] remains as [tex]\(\sqrt{7}\)[/tex] because 7 is a prime number.
- [tex]\(\sqrt{g^4} = g^2\)[/tex]
- [tex]\(\sqrt{h^8} = h^4\)[/tex]
Combine these results:
[tex]\[
2 \times 3 \times \sqrt{7} \times g^2 \times h^4
\][/tex]
4. Combine the coefficients:
Multiply the numerical coefficients together:
[tex]\[
2 \times 3 = 6
\][/tex]
So the simplified expression is:
[tex]\[
6 \sqrt{7} g^2 h^4
\][/tex]
Therefore, in the simplest form, the given expression is:
[tex]\[
6\sqrt{7}g^2h^4
\][/tex]
For your convenience, here is the expression without spaces or multiplication symbols between the coefficients or variables:
[tex]\[
\boxed{6sqrt{7}g^2h^4}
\][/tex]
