Example Question - conjugate method
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Rationalizing Denominators of Square Roots
To solve the given expression, we can use the technique of rationalizing the denominator. The expression given is: \[ \frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}} \] We will multiply both the numerator and denominator by the conjugate of the denominator to eliminate the square roots in the denominator. The conjugate of \(\sqrt{2} - \sqrt{3}\) is \(\sqrt{2} + \sqrt{3}\). Here's how it is done: \[ \frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}} \times \frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} + \sqrt{3}} = \frac{(\sqrt{2} + \sqrt{3})^2}{(\sqrt{2})^2 - (\sqrt{3})^2} \] Expanding the numerator and simplifying the denominator: \[ \frac{2 + 2\sqrt{2}\sqrt{3} + 3}{2 - 3} = \frac{5 + 2\sqrt{6}}{-1} \] Simplifying further: \[ \frac{5 + 2\sqrt{6}}{-1} = -5 - 2\sqrt{6} \] The final simplified result is: \[ -5 - 2\sqrt{6} \]
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: \[ q = -\frac{7}{3}, \quad r = -\frac{2}{3}, \quad s = 10 \]
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