If `3^(x+3)+3^(x-2)=244`, what is the value of x ?

Options

A. 2
B. 3
C. 1
D. 4

Answer

To solve the equation \( 3^{(x+3)} + 3^{(x-2)} = 244 \), we can follow these steps:

### Step 1: Rewrite the terms
We can rewrite the terms \( 3^{(x+3)} \) and \( 3^{(x-2)} \) using the properties of exponents:
\[ 3^{(x+3)} = 3^x \cdot 3^3 = 3^x \cdot 27 \]
\[ 3^{(x-2)} = 3^x \cdot 3^{-2} = 3^x \cdot \frac{1}{9} \]

### Step 2: Substitute back into the equation
Now, we substitute these rewritten terms back into the original equation:
\[ 3^x \cdot 27 + 3^x \cdot \frac{1}{9} = 244 \]

### Step 3: Factor out \( 3^x \)
Next, we can factor out \( 3^x \):
\[ 3^x \left( 27 + \frac{1}{9} \right) = 244 \]

### Step 4: Simplify the expression in parentheses
To simplify \( 27 + \frac{1}{9} \), we can convert 27 into a fraction:
\[ 27 = \frac{243}{9} \]
So,
\[ 27 + \frac{1}{9} = \frac{243}{9} + \frac{1}{9} = \frac{244}{9} \]

### Step 5: Substitute back into the equation
Now, we substitute this back into our equation:
\[ 3^x \cdot \frac{244}{9} = 244 \]

### Step 6: Eliminate the fraction
To eliminate the fraction, we can multiply both sides by 9:
\[ 3^x \cdot 244 = 244 \cdot 9 \]
This simplifies to:
\[ 3^x \cdot 244 = 2206 \]

### Step 7: Divide both sides by 244
Next, we divide both sides by 244:
\[ 3^x = \frac{2206}{244} \]
Calculating the right side gives:
\[ 3^x = 9 \]

### Step 8: Rewrite 9 as a power of 3
We know that \( 9 = 3^2 \), so we can write:
\[ 3^x = 3^2 \]

### Step 9: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ x = 2 \]

### Final Answer
Thus, the value of \( x \) is \( 2 \).

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