If `2 ^( x + 2) + 2 ^( x-1)= 36` then what is the value of x ?

Options

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

Explanation

To solve the equation \( 2^{(x + 2)} + 2^{(x - 1)} = 36 \), we can follow these steps:

### Step 1: Rewrite the exponents
We can rewrite \( 2^{(x + 2)} \) and \( 2^{(x - 1)} \) using the properties of exponents:
\[ 2^{(x + 2)} = 2^x \cdot 2^2 = 4 \cdot 2^x \]
\[ 2^{(x - 1)} = 2^x \cdot 2^{-1} = \frac{2^x}{2} \]

### Step 2: Substitute back into the equation
Substituting these back into the original equation gives us:
\[ 4 \cdot 2^x + \frac{2^x}{2} = 36 \]

### Step 3: Combine like terms
To combine the terms, we can factor out \( 2^x \):
\[ 2^x \left( 4 + \frac{1}{2} \right) = 36 \]

### Step 4: Find a common denominator
Finding a common denominator for \( 4 + \frac{1}{2} \):
\[ 4 = \frac{8}{2} \quad \text{so} \quad 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2} \]

### Step 5: Substitute back into the equation
Now we can substitute this back into the equation:
\[ 2^x \cdot \frac{9}{2} = 36 \]

### Step 6: Solve for \( 2^x \)
To isolate \( 2^x \), multiply both sides by \( \frac{2}{9} \):
\[ 2^x = 36 \cdot \frac{2}{9} \]
Calculating the right side:
\[ 2^x = 36 \cdot \frac{2}{9} = 8 \]

### Step 7: Express \( 8 \) as a power of \( 2 \)
We know that \( 8 = 2^3 \):
\[ 2^x = 2^3 \]

### Step 8: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ x = 3 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{3} \]