To rationalize the numerator of the expression \(\frac{1 + 4\sqrt{3}}{3\sqrt{5}}\), we will multiply both the numerator and the denominator by the conjugate of the numerator, which in this case is \(1 - 4\sqrt{3}\).
Step 1: Identify the Expression
\[ \frac{1 + 4\sqrt{3}}{3\sqrt{5}} \]
Step 2: Multiply by the Conjugate
Multiply the numerator and denominator by the conjugate \(1 - 4\sqrt{3}\): \[ \frac{(1 + 4\sqrt{3})(1 - 4\sqrt{3})}{(3\sqrt{5})(1 - 4\sqrt{3})} \]
Step 3: Simplify the Numerator
Using the difference of squares formula, which states that \((a + b)(a - b) = a^2 - b^2\): \[ (1 + 4\sqrt{3})(1 - 4\sqrt{3}) = 1^2 - (4\sqrt{3})^2 = 1 - 16 \times 3 = 1 - 48 = -47 \]
Step 4: Simplify the Denominator
Now, calculate the denominator: \[ 3\sqrt{5}(1 - 4\sqrt{3}) = 3\sqrt{5} - 12\sqrt{15} \]
Step 5: Combine the Results
The expression becomes: \[ \frac{-47}{3\sqrt{5} - 12\sqrt{15}} \]
Step 6: Final Expression
Thus, the rationalized form is: \[ \frac{-47}{3\sqrt{5} - 12\sqrt{15}} \]
This matches up with one of the options you provided. The final result is: \[ -\frac{47}{3\sqrt{5} - 12\sqrt{15}} \]
This corresponds to the option: \[ -\frac{47}{35\sqrt{} - 12\sqrt{15}} \] where it looks like there may have been some simplification intended in your original options. If needed, please clarify which initial problem you'd like simplified further!