If `(5/2)^(5)xx(5/2)^(x+3)=(2/5)^(-10)` then the value of x is

Options

A. `-1`
B. `0`
C. `10`
D. `2`

Answer

To solve the equation \((\frac{5}{2})^{5} \times (\frac{5}{2})^{(x+3)} = (\frac{2}{5})^{-10}\), we can follow these steps:

### Step 1: Combine the Left Side
Using the property of exponents that states \(a^m \times a^n = a^{m+n}\), we can combine the left side of the equation:

\[ (\frac{5}{2})^{5 + (x + 3)} = (\frac{5}{2})^{(x + 8)} \]

### Step 2: Rewrite the Right Side
Next, we need to rewrite the right side of the equation. We know that \((\frac{2}{5})^{-10}\) can be rewritten using the property of exponents \(\frac{1}{a^{-n}} = a^{n}\):

\[ (\frac{2}{5})^{-10} = (\frac{5}{2})^{10} \]

### Step 3: Set the Exponents Equal
Now that both sides of the equation have the same base \((\frac{5}{2})\), we can set the exponents equal to each other:

\[ x + 8 = 10 \]

### Step 4: Solve for \(x\)
To find the value of \(x\), we can subtract 8 from both sides:

\[ x = 10 - 8 \]
\[ x = 2 \]

### Final Answer
The value of \(x\) is \(2\).

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