1.

Simplify -​

Answer» EN Expression:

\sf{\dfrac{3}{\sqrt{5}+\sqrt{2}}-\dfrac{2}{\sqrt{7}+\sqrt{5}}+\dfrac{5}{\sqrt{7}-\sqrt{2}}

Solution:

In order to simplify the expression, we need to RATIONALIZE each term:

\sf{=\dfrac{3}{\sqrt{5}+\sqrt{2}}\times\dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}-\dfrac{2}{\sqrt{7}+\sqrt{5}}\times\dfrac{\sqrt{7}-\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\dfrac{5}{\sqrt{7}-\sqrt{2}}\times\dfrac{\sqrt{7}+\sqrt{2}}{\sqrt{7}+\sqrt{2}}

\sf{=\dfrac{3\sqrt{5}-3\sqrt{2}}{5-2}-\dfrac{2\sqrt{7}-2\sqrt{5}}{7-5}+\dfrac{5\sqrt{7}+5\sqrt{2}}{7-2}}

\sf{=\dfrac{3(\sqrt{5}-\sqrt{2})}{3}-\dfrac{2(\sqrt{7}-\sqrt{5})}{2}+\dfrac{(\sqrt{7}+\sqrt{2})}{5}}

Cancelling the like terms, we are LEFT with:

\sf{=\sqrt{5}-\sqrt{2}-\sqrt{7}+\sqrt{5}+\sqrt{7}+\sqrt{2}}

Cancelling the like terms with negative signs:

\sf{=\sqrt{5}+\sqrt{5}}

\bf{=2\sqrt{5}}

Hence the simplified answer is 2√5.

Rationalizing the denominator:

To rationalize the denominator of a FRACTION, we are REMOVING the rational numbers from the denominator by multiply it with the terms of denominator with the opposite sign.


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