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Write this expression in simplest form.

[tex]\sqrt{252 g^4 h^8}[/tex]

Don't include any spaces or multiplication symbols between coefficients or variables in your answer.


Sagot :

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]