Notice the pattern in both quantities. They are of the form:


\(\frac{x+\sqrt{y}}{(x-\sqrt{y})^n}\)


where \(x^2-y=1\) in both cases:


For Quantity A: \(4^2-15=1 \)
For Quantity B: \( 6^2-35=1 \)


Use the identity:


\(\frac{x+\sqrt{y}}{x-\sqrt{y}}=x+\sqrt{y}\)


because \((x-\sqrt{y})(x+\sqrt{y})=x^2-y=1\) .

So for Quantity A:


\(\frac{4+\sqrt{15}}{4-\sqrt{15}}=4+\sqrt{15}\)


Thus:


\(\frac{4+\sqrt{15}}{(4-\sqrt{15})^3}=(4+\sqrt{15})^2\)


For Quantity B:


\(\frac{6+\sqrt{35}}{6-\sqrt{35}}=6+\sqrt{35}\)


Thus:


\(\frac{6+\sqrt{35}}{(6-\sqrt{35})^2}=6+\sqrt{35}\)


Now compare:


\((4+\sqrt{15})^2 \text { vs. } 6+\sqrt{35}\)


Approximate:



\(4+\sqrt{15} \approx 4+3.873=7.873 \)

\((7.873)^2 \approx 62.0\)



Also:


\(6+\sqrt{35} \approx 6+5.916=11.916\)


Clearly Quantity A is much larger.

A) Quantity A is greater.