If `10^(k)=64`, what is the value of `10^((k)/(2)+1)`?

Options

A. `18`
B. `42`
C. `80`
D. `81`

Answer

To solve the equation \( 10^k = 64 \) and find the value of \( 10^{\left(\frac{k}{2} + 1\right)} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ 10^k = 64 \]

### Step 2: Take the square root of both sides
Taking the square root of both sides gives us:
\[ \sqrt{10^k} = \sqrt{64} \]

### Step 3: Simplify the square root
We can express the left side as:
\[ 10^{\frac{k}{2}} = 8 \]
because \( \sqrt{64} = 8 \).

### Step 4: Multiply both sides by \( 10 \)
Now, we multiply both sides of the equation by \( 10 \):
\[ 10^{\frac{k}{2}} \cdot 10 = 8 \cdot 10 \]

### Step 5: Use the property of exponents
Using the property of exponents \( a^m \cdot a^n = a^{m+n} \), we can rewrite the left side:
\[ 10^{\frac{k}{2} + 1} = 80 \]

### Conclusion
Thus, the value of \( 10^{\left(\frac{k}{2} + 1\right)} \) is:
\[ 80 \]

### Final Answer
The value of \( 10^{\left(\frac{k}{2} + 1\right)} \) is \( 80 \).

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