How do you simplify $\frac{2 \sqrt{5}}{\sqrt{2} + 5}$ $(2sqrt5)/ (sqrt2 +5)$ ?

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How do you simplify $\frac{2 \sqrt{5}}{\sqrt{2} + 5}$ $(2sqrt5)/ (sqrt2 +5)$ ?

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Answer 1 · 2015-04-09T08:36:11

To simplify the expression, we need to make it's Denominator RATIONAL. And to do that, we multiply the numerator and the denominator with it's CONJUGATE.

The denominator is $\sqrt{2} + 5$ $sqrt2 + 5$ which is the same as $5 + \sqrt{2}$ $5 + sqrt 2$

and it's CONJUGATE is $5 - \sqrt{2}$ $5 - sqrt2$

$\frac{2 \sqrt{5}}{5 + \sqrt{2}} \cdot \frac{5 - \sqrt{2}}{5 - \sqrt{2}}$ $(2sqrt5)/ (5+sqrt2) * (5 - sqrt 2)/(5 - sqrt 2)$ (This keeps the Ratio unchanged)

$= \frac{(2 \sqrt{5}) \cdot (5 - \sqrt{2})}{5^{2} - {(\sqrt{2})}^{2}}$ $= ((2sqrt5) * (5 - sqrt 2))/(5^2 - (sqrt 2)^2)$

$= \frac{(2 \sqrt{5}) \cdot (5 - \sqrt{2})}{25 - 2}$ $= ((2sqrt5) * (5 - sqrt 2))/(25 - 2)$

$= \frac{(2 \sqrt{5}) \cdot (5 - \sqrt{2})}{23}$ $= ((2sqrt5) * (5 - sqrt 2))/(23)$

$= \frac{10 \sqrt{5} - 2 \sqrt{10}}{23}$ $= (10sqrt5 - 2sqrt 10)/(23)$

As the denominator 23 is RATIONAL, we have successfully simplified the expression.

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How do you simplify $\frac{2 \sqrt{5}}{\sqrt{2} + 5}$ $(2sqrt5)/ (sqrt2 +5)$ ?

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