What is the value of x, if `2^(x+5)= 512`?

Answer

To solve the equation \(2^{(x+5)} = 512\), we will follow these steps:

### Step 1: Rewrite 512 as a power of 2
First, we need to express 512 as a power of 2. We can do this by repeatedly dividing 512 by 2 until we reach 1.

- \(512 \div 2 = 256\)
- \(256 \div 2 = 128\)
- \(128 \div 2 = 64\)
- \(64 \div 2 = 32\)
- \(32 \div 2 = 16\)
- \(16 \div 2 = 8\)
- \(8 \div 2 = 4\)
- \(4 \div 2 = 2\)
- \(2 \div 2 = 1\)

Counting the divisions, we see that we divided by 2 a total of 9 times. Therefore, we can write:
\[ 512 = 2^9 \]

### Step 2: Set the exponents equal to each other
Now that we have \(512\) expressed as \(2^9\), we can rewrite the original equation:
\[ 2^{(x+5)} = 2^9 \]

Since the bases are the same, we can set the exponents equal to each other:
\[ x + 5 = 9 \]

### Step 3: Solve for x
Now, we will solve for \(x\) by isolating it:
\[ x = 9 - 5 \]
\[ x = 4 \]

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

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