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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]
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]
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