यदि `25 ^(-2x) = ( 5 ^((48)/(2)))/(5 ^((26)/(2)). 25 ^((19)/(2))`है तब x = ?

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the equation \( 25^{-2x} = \frac{5^{(48/2)}}{5^{(26/2)} \cdot 25^{(19/2)}} \), we will follow these steps:

### Step 1: Rewrite 25 in terms of 5
We know that \( 25 = 5^2 \). Therefore, we can rewrite \( 25^{-2x} \) as:
\[ (5^2)^{-2x} = 5^{-4x} \]

### Step 2: Simplify the right side of the equation
Now, let's simplify the right side:
\[ \frac{5^{(48/2)}}{5^{(26/2)} \cdot 25^{(19/2)}} \]
Calculating the exponents:
\[ \frac{5^{24}}{5^{13} \cdot (5^2)^{(19/2)}} \]
Now, simplify \( (5^2)^{(19/2)} \):
\[ (5^2)^{(19/2)} = 5^{19} \]
So, the right side becomes:
\[ \frac{5^{24}}{5^{13} \cdot 5^{19}} = \frac{5^{24}}{5^{32}} = 5^{24 - 32} = 5^{-8} \]

### Step 3: Set the exponents equal to each other
Now we have:
\[ 5^{-4x} = 5^{-8} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ -4x = -8 \]

### Step 4: Solve for x
Now, divide both sides by -4:
\[ x = \frac{-8}{-4} = 2 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{2} \]