`sqrt7div7sqrt7=?`

Options

A. 7
B. `(1)/(7)`
C. `(1)/(sqrt7)`
D. `sqrt(7)`

Explanation

To solve the problem \( \frac{\sqrt{7}}{7\sqrt{7}} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{\sqrt{7}}{7\sqrt{7}} \]

### Step 2: Simplify the denominator
We can express the denominator as:
\[ 7\sqrt{7} = 7 \cdot 7^{1/2} = 7^{1 + 1/2} = 7^{3/2} \]
Thus, the expression becomes:
\[ \frac{\sqrt{7}}{7^{3/2}} \]

### Step 3: Rewrite the numerator
The numerator can also be expressed as:
\[ \sqrt{7} = 7^{1/2} \]
So now we have:
\[ \frac{7^{1/2}}{7^{3/2}} \]

### Step 4: Apply the properties of exponents
Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify:
\[ \frac{7^{1/2}}{7^{3/2}} = 7^{1/2 - 3/2} = 7^{-2/2} = 7^{-1} = \frac{1}{7} \]

### Conclusion
Thus, the final answer is:
\[ \frac{\sqrt{7}}{7\sqrt{7}} = \frac{1}{7} \]

### Answer
The answer is \( \frac{1}{7} \), which corresponds to option B.

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