Which expression is equivalent to [tex]\sqrt{\frac{900 f^6}{100 g^4}}[/tex]?

A. [tex]\frac{3\left|f^3\right|}{g^2}[/tex]
B. [tex]\frac{3\left|f^3\right|}{g^2}[/tex] (duplicate option, should be removed if present in the original)
C. [tex]\frac{30\left|f^3\right|}{g^2}[/tex]
D. [tex]30\left|f^3\right| g^2[/tex]



Answer :

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]