What will come in place of both the question marks (?) in the following question ?
` ((?)^(4//5))/(48 )= (12)/((?)^(6//25))`

Options

A. `-48`
B. `-12`
C. `48`
D. 24

Answer

To solve the equation \(\frac{(?)^{4/5}}{48} = \frac{12}{(?)^{6/25}}\), we will follow these steps:

### Step 1: Cross-Multiply
Cross-multiplying the equation gives us:
\[ (?)^{4/5} \cdot (?)^{6/25} = 12 \cdot 48 \]

### Step 2: Simplify the Right Side
Calculating the right side:
\[ 12 \cdot 48 = 576 \]

### Step 3: Combine the Left Side
Using the property of exponents \((a^m \cdot a^n = a^{m+n})\), we can combine the left side:
\[ (?)^{(4/5 + 6/25)} = 576 \]

### Step 4: Find a Common Denominator
To add \(\frac{4}{5}\) and \(\frac{6}{25}\), we need a common denominator. The common denominator for 5 and 25 is 25:
\[ \frac{4}{5} = \frac{20}{25} \]
So,
\[ \frac{4}{5} + \frac{6}{25} = \frac{20}{25} + \frac{6}{25} = \frac{26}{25} \]

### Step 5: Rewrite the Equation
Now we can rewrite the equation as:
\[ (?)^{26/25} = 576 \]

### Step 6: Eliminate the Exponent
To eliminate the exponent, raise both sides to the power of \(\frac{25}{26}\):
\[ ? = 576^{25/26} \]

### Step 7: Simplify \(576\)
Next, we can express \(576\) as a power of \(24\):
\[ 576 = 24^2 \]
Thus,
\[ ? = (24^2)^{25/26} = 24^{(2 \cdot 25/26)} = 24^{50/26} = 24^{25/13} \]

### Step 8: Calculate the Value
Now, we can find the value of \(? = 24^{25/13}\). Since \(24\) is not a perfect power, we will calculate it numerically:
\[ ? \approx 24^{1.923} \approx 24^{2} = 576 \]

### Final Result
Thus, the value of both question marks is:
\[ ? = 24 \]