If `9 ^(x +2) = 240 + 9 ^(x) ,` then value of `(8x) ^(x)` is

Options

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

Answer

To solve the equation \( 9^{(x + 2)} = 240 + 9^x \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 9^{(x + 2)} = 240 + 9^x \]
We can rewrite \( 9^{(x + 2)} \) as:
\[ 9^{(x + 2)} = 9^x \cdot 9^2 = 9^x \cdot 81 \]
So, the equation becomes:
\[ 9^x \cdot 81 = 240 + 9^x \]

### Step 2: Move all terms involving \( 9^x \) to one side
Next, we can rearrange the equation:
\[ 9^x \cdot 81 - 9^x = 240 \]
Factoring out \( 9^x \):
\[ 9^x (81 - 1) = 240 \]
This simplifies to:
\[ 9^x \cdot 80 = 240 \]

### Step 3: Solve for \( 9^x \)
Now, we can isolate \( 9^x \):
\[ 9^x = \frac{240}{80} = 3 \]

### Step 4: Solve for \( x \)
Since \( 9^x = 3 \), we can express 9 as \( 3^2 \):
\[ (3^2)^x = 3 \]
This simplifies to:
\[ 3^{2x} = 3^1 \]
By equating the exponents, we have:
\[ 2x = 1 \]
Thus, solving for \( x \):
\[ x = \frac{1}{2} \]

### Step 5: Calculate \( (8x)^x \)
Now we need to find the value of \( (8x)^x \):
\[ (8x)^x = (8 \cdot \frac{1}{2})^{\frac{1}{2}} = 4^{\frac{1}{2}} = \sqrt{4} = 2 \]

### Final Answer
The value of \( (8x)^x \) is:
\[ \boxed{2} \]