If `8^(13) = 2^(z)`, then z =

Options

A. 19
B. 13
C. 10
D. 39

Answer

To solve the equation \( 8^{13} = 2^{z} \), we can follow these steps:

### Step 1: Rewrite 8 as a power of 2
We know that \( 8 \) can be expressed as \( 2^3 \). Therefore, we can rewrite the left side of the equation:
\[ 8^{13} = (2^3)^{13} \]

### Step 2: Apply the power of a power property
Using the property of exponents that states \( (a^m)^n = a^{m \cdot n} \), we can simplify the left side:
\[ (2^3)^{13} = 2^{3 \cdot 13} \]

### Step 3: Calculate the exponent
Now, we calculate \( 3 \cdot 13 \):
\[ 3 \cdot 13 = 39 \]
So, we have:
\[ 2^{39} \]

### Step 4: Set the exponents equal
Now we can rewrite the equation:
\[ 2^{39} = 2^{z} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ 39 = z \]

### Step 5: Conclusion
Thus, the value of \( z \) is:
\[ z = 39 \]