If `7^(k)=100`, what is the value of `7^((k)/(2)+1)`?

Options

A. `18`
B. `51`
C. `57`
D. `70`

Answer

To solve the problem, we need to find the value of \( 7^{\left(\frac{k}{2} + 1\right)} \) given that \( 7^k = 100 \).

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression we want to evaluate:
\[ 7^{\left(\frac{k}{2} + 1\right)} \]
This can be rewritten using the property of exponents:
\[ 7^{\left(\frac{k}{2} + 1\right)} = 7^{\frac{k}{2}} \cdot 7^1 \]

2. **Substitute \( 7^1 \)**:
Since \( 7^1 = 7 \), we can express the equation as:
\[ 7^{\left(\frac{k}{2} + 1\right)} = 7^{\frac{k}{2}} \cdot 7 \]

3. **Evaluate \( 7^{\frac{k}{2}} \)**:
To find \( 7^{\frac{k}{2}} \), we can use the property of exponents that states:
\[ 7^{\frac{k}{2}} = \sqrt{7^k} \]
Since we know \( 7^k = 100 \), we can substitute this into our equation:
\[ 7^{\frac{k}{2}} = \sqrt{100} \]

4. **Calculate the Square Root**:
The square root of 100 is:
\[ \sqrt{100} = 10 \]

5. **Combine the Results**:
Now we can substitute back into our expression:
\[ 7^{\left(\frac{k}{2} + 1\right)} = 10 \cdot 7 \]

6. **Final Calculation**:
Finally, we calculate:
\[ 10 \cdot 7 = 70 \]

Thus, the value of \( 7^{\left(\frac{k}{2} + 1\right)} \) is \( \boxed{70} \).