To solve the problem of identifying which choice is equivalent to the given fraction [tex]\(\frac{7}{7+\sqrt{14x}}\)[/tex], we need to rationalize the denominator. Rationalizing the denominator means eliminating the square root from the denominator by multiplying it by a conjugate expression. The conjugate of [tex]\(7 + \sqrt{14x}\)[/tex] is [tex]\(7 - \sqrt{14x}\)[/tex].
Let's perform the following steps to rationalize the denominator and simplify the expression:
1. Identify the conjugate of the denominator:
The denominator is [tex]\(7 + \sqrt{14x}\)[/tex].
The conjugate of [tex]\(7 + \sqrt{14x}\)[/tex] is [tex]\(7 - \sqrt{14x}\)[/tex].
2. Multiply the numerator and denominator by the conjugate of the denominator:
[tex]\[
\frac{7}{7 + \sqrt{14x}} \times \frac{7 - \sqrt{14x}}{7 - \sqrt{14x}}
\][/tex]
3. Perform the multiplication in the numerator:
[tex]\[
7 \times (7 - \sqrt{14x}) = 7 \cdot 7 - 7 \cdot \sqrt{14x} = 49 - 7\sqrt{14x}
\][/tex]
4. Perform the multiplication in the denominator:
[tex]\[
(7 + \sqrt{14x})(7 - \sqrt{14x}) = 7^2 - (\sqrt{14x})^2 = 49 - 14x
\][/tex]
5. Combine the results:
[tex]\[
\frac{7(7 - \sqrt{14x})}{(7 + \sqrt{14x})(7 - \sqrt{14x})} = \frac{49 - 7\sqrt{14x}}{49 - 14x}
\][/tex]
Thus, the fraction [tex]\(\frac{7}{7 + \sqrt{14x}}\)[/tex] simplifies to [tex]\(\frac{7 - \sqrt{14x}}{49 - 14x}\)[/tex].
Now, let's compare this result with the provided choices:
- A. [tex]\(\frac{7 - \sqrt{14x}}{49 - 2x}\)[/tex]
- B. [tex]\(\frac{7 - \sqrt{14x}}{7 - 2x}\)[/tex]
- C. [tex]\(\frac{7 - \sqrt{14x}}{7 - 14x}\)[/tex]
- D. [tex]\(\frac{7 - \sqrt{14x}}{49 - 14x}\)[/tex]
The expression [tex]\(\frac{7 - \sqrt{14x}}{49 - 14x}\)[/tex] matches option D.
So, the correct choice is:
D. [tex]\(\frac{7 - \sqrt{14x}}{49 - 14x}\)[/tex]
