Rationalizing the Denominator of a Complex Fraction
To express the given expression in the form \( q + r \sqrt{s} \), where \( q, r, \) and \( s \) are rational numbers, we need to rationalize the denominator. The given expression is:
\[ \frac{2\sqrt{5} + 5\sqrt{2}}{2\sqrt{5} - 5\sqrt{2}} \]
First, we can multiply the numerator and the denominator by the conjugate of the denominator to eliminate the square roots in the denominator. The conjugate of \(2\sqrt{5} - 5\sqrt{2}\) is \(2\sqrt{5} + 5\sqrt{2}\). Multiplying both the numerator and the denominator by the conjugate, we get:
\[ \frac{(2\sqrt{5} + 5\sqrt{2})(2\sqrt{5} + 5\sqrt{2})}{(2\sqrt{5} - 5\sqrt{2})(2\sqrt{5} + 5\sqrt{2})} \]
Now expand both the numerator and the denominator:
The numerator, when multiplied out, becomes:
\[ (2\sqrt{5} \cdot 2\sqrt{5}) + (2\sqrt{5} \cdot 5\sqrt{2}) + (5\sqrt{2} \cdot 2\sqrt{5}) + (5\sqrt{2} \cdot 5\sqrt{2}) \]
\[ = 4 \cdot 5 + 2 \cdot 5 \cdot \sqrt{10} + 5 \cdot 2 \cdot \sqrt{10} + 25 \cdot 2 \]
\[ = 20 + 10\sqrt{10} + 10\sqrt{10} + 50 \]
\[ = 70 + 20\sqrt{10} \]
The denominator, when multiplied out, becomes a difference of squares:
\[ (2\sqrt{5})^2 - (5\sqrt{2})^2 \]
\[ = 4 \cdot 5 - 25 \cdot 2 \]
\[ = 20 - 50 \]
\[ = -30 \]
Combining the expanded numerator and the expanded denominator, we have:
\[ \frac{70 + 20\sqrt{10}}{-30} \]
Now, simplify this by dividing both terms in the numerator by the denominator:
\[ \frac{70}{-30} + \frac{20\sqrt{10}}{-30} \]
\[ = -\frac{7}{3} - \frac{2\sqrt{10}}{3} \]
So the final answer, expressed in the form \( q + r \sqrt{s} \), is:
\[ q = -\frac{7}{3}, \quad r = -\frac{2}{3}, \quad s = 10 \]
