To determine which expression is equivalent to [tex]\( \sqrt{\frac{900 f^6}{100 g^4}} \)[/tex], let’s go through the simplification steps step-by-step:
### Step 1: Simplify the Fraction Inside the Square Root
First, consider the fraction inside the square root:
[tex]\[
\frac{900 f^6}{100 g^4}
\][/tex]
We can simplify the numerical part (coefficients):
[tex]\[
\frac{900}{100} = 9
\][/tex]
So the expression becomes:
[tex]\[
\sqrt{9 \cdot \frac{f^6}{g^4}}
\][/tex]
### Step 2: Split the Square Root
We can now split the square root into two separate square roots:
[tex]\[
\sqrt{9} \cdot \sqrt{\frac{f^6}{g^4}}
\][/tex]
### Step 3: Evaluate the Square Root of the Numerical Part
[tex]\[
\sqrt{9} = 3
\][/tex]
So the expression updates to:
[tex]\[
3 \cdot \sqrt{\frac{f^6}{g^4}}
\][/tex]
### Step 4: Simplify the Algebraic Part
Next, we simplify the square root of the algebraic part:
[tex]\[
\sqrt{\frac{f^6}{g^4}} = \frac{\sqrt{f^6}}{\sqrt{g^4}}
\][/tex]
Since:
[tex]\[
\sqrt{f^6} = f^3 \quad \text{and} \quad \sqrt{g^4} = g^2
\][/tex]
We get:
[tex]\[
\frac{\sqrt{f^6}}{\sqrt{g^4}} = \frac{f^3}{g^2}
\][/tex]
### Step 5: Combine the Results
Bringing it all together, we have:
[tex]\[
3 \cdot \frac{f^3}{g^2} = \frac{3 f^3}{g^2}
\][/tex]
Now considering the absolute value:
[tex]\[
\frac{3 \left| f^3 \right|}{g^2}
\][/tex]
Thus, the expression equivalent to [tex]\( \sqrt{\frac{900 f^6}{100 g^4}} \)[/tex] is:
[tex]\[
\frac{3 \left| f^3 \right|}{g^2}
\][/tex]
### Conclusion
The correct answer is:
[tex]\[
\boxed{\frac{3 \left| f^3 \right|}{g^2}}
\][/tex]
