How do you rationalize the denominator $\frac{2}{5 - \sqrt{3}}$ $2/(5-sqrt3)$ ?

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How do you rationalize the denominator $\frac{2}{5 - \sqrt{3}}$ $2/(5-sqrt3)$ ?

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Answer 1 · 2015-06-03T08:50:49

$\frac{2}{5 - \sqrt{3}}$ $2/(5-sqrt3)$

We can ratioinalise the denominator by multiplying the numerator and denominator of the expression by the conjugate of $5 - \sqrt{3}$ $5-sqrt3$

The conjugate of $5 - \sqrt{3} = 5 + \sqrt{3}$ $5-sqrt3 = color(red)(5+sqrt3$

$= \frac{2 \times (5 + \sqrt{3})}{(5 - \sqrt{3}) \times (5 + \sqrt{3})}$ $=(2xxcolor(red)((5+sqrt3)))/((5-sqrt3)xxcolor(red)((5+sqrt3))$

We know that $(a - b) (a + b) = a^{2} - b^{2}$ $color(blue)((a-b)(a+b)=a^2 - b^2$
so , $(5 - \sqrt{3}) \times (5 + \sqrt{3}) = 5^{2} - {\sqrt{3}}^{2} = 25 - 3 = 22$ $(5-sqrt3)xx(5+sqrt3) = 5^2- sqrt3^2 = color(blue)(25 - 3 = 22$

the expression becomes:
$(cancel2xxcolor(red)((5+sqrt3)))/cancel22$

$= (5+ sqrt3)/ 11$

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