Understanding Rationalization
Rationalizing the denominator is a technique used to eliminate irrational numbers from the denominator of a fraction. In this case, we have the expression:
\[
\frac{\sqrt{7} - 5}{\sqrt{7} + 5}
\]
To rationalize, we will multiply both the numerator and the denominator by the conjugate of the denominator.
Step 1: Identify the Conjugate
- The conjugate of \(\sqrt{7} + 5\) is \(\sqrt{7} - 5\).
Step 2: Multiply by the Conjugate
- Multiply both the numerator and denominator by \(\sqrt{7} - 5\):
\[
\frac{(\sqrt{7} - 5)(\sqrt{7} - 5)}{(\sqrt{7} + 5)(\sqrt{7} - 5)}
\]
Step 3: Simplify the Denominator
- Utilize the difference of squares formula:
\[
(\sqrt{7})^2 - (5)^2 = 7 - 25 = -18
\]
Step 4: Simplify the Numerator
- Expand the numerator:
\[
(\sqrt{7} - 5)(\sqrt{7} - 5) = (\sqrt{7})^2 - 2(\sqrt{7})(5) + (5)^2 = 7 - 10\sqrt{7} + 25 = 32 - 10\sqrt{7}
\]
Final Expression
- Combine the results:
\[
\frac{32 - 10\sqrt{7}}{-18}
\]
- This can be simplified further:
\[
\frac{-32 + 10\sqrt{7}}{18}
\]
- Or:
\[
\frac{10\sqrt{7} - 32}{18}
\]
Conclusion
- The rationalized form of \(\frac{\sqrt{7} - 5}{\sqrt{7} + 5}\) is \(\frac{10\sqrt{7} - 32}{18}\). This process eliminates the square root from the denominator, making the expression neater and easier to work with.