If `3^(x)+3^(x+1)=36`, then the value of `x^(x)` is :

Options

A. 64
B. 3125
C. 81
D. 4

Answer

To solve the equation \(3^x + 3^{x+1} = 36\), we can follow these steps:

### Step 1: Rewrite the equation
The equation can be rewritten as:
\[ 3^x + 3^{x+1} = 36 \]
We know that \(3^{x+1} = 3^x \cdot 3\). Therefore, we can rewrite the equation as:
\[ 3^x + 3 \cdot 3^x = 36 \]

### Step 2: Factor out \(3^x\)
Now, we can factor out \(3^x\):
\[ 3^x (1 + 3) = 36 \]
This simplifies to:
\[ 3^x \cdot 4 = 36 \]

### Step 3: Solve for \(3^x\)
Next, we can isolate \(3^x\) by dividing both sides by 4:
\[ 3^x = \frac{36}{4} \]
This simplifies to:
\[ 3^x = 9 \]

### Step 4: Express 9 as a power of 3
We know that \(9\) can be expressed as \(3^2\):
\[ 3^x = 3^2 \]

### Step 5: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ x = 2 \]

### Step 6: Calculate \(x^x\)
Now that we have found \(x\), we can calculate \(x^x\):
\[ x^x = 2^2 = 4 \]

### Final Answer
Thus, the value of \(x^x\) is:
\[ \boxed{4} \]

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