Sure, let's break down the given expression step-by-step to find its equivalent form:
Given expression:
[tex]\[
\sqrt{\frac{900 f^6}{100 g^4}}
\][/tex]
Step 1: Simplify the fraction inside the square root.
We start by simplifying the fraction [tex]\(\frac{900}{100}\)[/tex]:
[tex]\[
\frac{900 f^6}{100 g^4} = \frac{900}{100} \cdot \frac{f^6}{g^4} = 9 \cdot \frac{f^6}{g^4}
\][/tex]
So the expression becomes:
[tex]\[
\sqrt{9 \cdot \frac{f^6}{g^4}}
\][/tex]
Step 2: Apply the square root to each part.
The square root of a product is the product of the square roots. Therefore:
[tex]\[
\sqrt{9 \cdot \frac{f^6}{g^4}} = \sqrt{9} \cdot \sqrt{\frac{f^6}{g^4}}
\][/tex]
Step 3: Calculate the individual square roots.
First, we take the square root of 9:
[tex]\[
\sqrt{9} = 3
\][/tex]
Next, we apply the square root to the remaining fraction:
[tex]\[
\sqrt{\frac{f^6}{g^4}} = \frac{\sqrt{f^6}}{\sqrt{g^4}}
\][/tex]
Step 4: Simplify the square roots of the individual variables.
- [tex]\(\sqrt{f^6}\)[/tex]: Recall that [tex]\(\sqrt{x^2} = |x|\)[/tex], so [tex]\(\sqrt{f^6} = f^{6/2} = f^3\)[/tex]
- [tex]\(\sqrt{g^4}\)[/tex]: Similarly, [tex]\(\sqrt{g^4} = g^{4/2} = g^2\)[/tex]
Thus, we have:
[tex]\[
\sqrt{\frac{f^6}{g^4}} = \frac{f^3}{g^2}
\][/tex]
Step 5: Combine all parts.
Putting it all back together:
[tex]\[
3 \cdot \frac{f^3}{g^2} = \frac{3 f^3}{g^2}
\][/tex]
Consider the absolute value.
Since we are dealing with square roots, which always yield non-negative results:
[tex]\[
f^3 \rightarrow |f^3|
\][/tex]
Final Expression:
[tex]\[
\sqrt{\frac{900 f^6}{100 g^4}} = \frac{3 |f^3|}{g^2}
\][/tex]
Thus, the equivalent expression is:
[tex]\[
\boxed{\frac{3 \left|f^3\right|}{g^2}}
\][/tex]
