For all def `ne 0, (4d^(3)e^(4)f^(2))/((2de^(3)f)^(2))=`

Options

A. `2de^(2)`
B. `2de^(-2)`
C. `de^(2)`
D. `de^(-2)`

Explanation

To solve the expression \( \frac{4d^{3}e^{4}f^{2}}{(2de^{3}f)^{2}} \), we will simplify it step by step.

### Step 1: Expand the Denominator
The denominator is \( (2de^{3}f)^{2} \). We can expand this using the property of exponents:
\[ (2de^{3}f)^{2} = (2^{2})(d^{2})(e^{3 \cdot 2})(f^{2}) = 4d^{2}e^{6}f^{2} \]

### Step 2: Rewrite the Expression
Now, we can rewrite the original expression with the expanded denominator:
\[ \frac{4d^{3}e^{4}f^{2}}{4d^{2}e^{6}f^{2}} \]

### Step 3: Cancel Common Factors
Next, we can cancel the common factors in the numerator and the denominator:
- The \( 4 \) in the numerator and denominator cancels out.
- The \( f^{2} \) in the numerator and denominator cancels out.

This simplifies our expression to:
\[ \frac{d^{3}e^{4}}{d^{2}e^{6}} \]

### Step 4: Apply the Quotient Rule for Exponents
Now we can simplify the expression further using the quotient rule for exponents:
\[ \frac{d^{3}}{d^{2}} = d^{3-2} = d^{1} = d \]
\[ \frac{e^{4}}{e^{6}} = e^{4-6} = e^{-2} \]

### Step 5: Combine the Results
Putting it all together, we have:
\[ d \cdot e^{-2} = \frac{d}{e^{2}} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{d}{e^{2}} \]