Question - Simplifying an Algebraic Expression
Solution:
The image shows an algebraic expression that needs to be simplified:\[ \frac{1 - \sqrt{2}}{\sqrt{5} - \sqrt{3}} + \frac{1 + \sqrt{2}}{\sqrt{5} + \sqrt{3}} \]To simplify this expression, we can combine the two fractions into a single fraction by finding a common denominator, which is $$(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})$$.First, we recognize that $$(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})$$ equals $$\sqrt{5}^2 - \sqrt{3}^2$$, which simplifies to $$5 - 3$$, or $$2$$.Now, let's work out the combined numerator:For denominator $$\sqrt{5} - \sqrt{3}$$, the corresponding numerator is $$(1 - \sqrt{2})(\sqrt{5} + \sqrt{3})$$.For denominator $$\sqrt{5} + \sqrt{3}$$, the corresponding numerator is $$(1 + \sqrt{2})(\sqrt{5} - \sqrt{3})$$.Let's find each product for the numerators separately:- $$(1 - \sqrt{2})(\sqrt{5} + \sqrt{3}) = \sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}$$- $$(1 + \sqrt{2})(\sqrt{5} - \sqrt{3}) = \sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6}$$Next, combine these to get the new numerator:\[ (\sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}) + (\sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6}) \]Simplify by cancelling out the opposite terms $$ \sqrt{3} $$ and $$ -\sqrt{3} $$, and by combining like terms:\[ 2\sqrt{5} - 2\sqrt{6} \]With the denominator found to be $$2$$, we now have:\[ \frac{2\sqrt{5} - 2\sqrt{6}}{2} \]Divide both terms in the numerator by the denominator to simplify further:\[ \sqrt{5} - \sqrt{6} \]Thus, the simplified expression is:\[ \sqrt{5} - \sqrt{6} \]
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