How do you rationalize the denominator and simplify $\frac{5}{\sqrt{3} - 1}$ $5/(sqrt3-1)$ ?

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How do you rationalize the denominator and simplify $\frac{5}{\sqrt{3} - 1}$ $5/(sqrt3-1)$ ?

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Answer 1 · 2016-04-15T14:12:47

$= \frac{5 \sqrt{3} + 5}{2}$ $=(5sqrt 3 + 5 ) / 2$

Explanation:

$\frac{5}{\sqrt{3} - 1}$ $5 / (sqrt3 -1 )$

To rationalize the expression, we multiply it by the conjugate of the denominator $(\sqrt{3} + 1)$ $color(blue)((sqrt 3 + 1 ))$

$\frac{5 \cdot (\sqrt{3} + 1)}{(\sqrt{3} - 1) \cdot (\sqrt{3} + 1)}$ $(5 * color(blue)((sqrt 3 + 1 ) ))/ ((sqrt3 -1 ) * color(blue)((sqrt 3 + 1 ))$

$= \frac{(5 \cdot (\sqrt{3}) + 5 \cdot (1))}{(\sqrt{3} - 1) \cdot (\sqrt{3} + 1)}$ $=((5 * color(blue)((sqrt 3)) + 5 * color(blue)(( 1 )) ))/ ((sqrt3 -1 ) * color(blue)((sqrt 3 + 1 ))$

$= \frac{5 \sqrt{3} + 5}{(\sqrt{3} - 1) \cdot (\sqrt{3} + 1)}$ $=(5sqrt 3 + 5 ) / ((sqrt3 -1 ) * color(blue)((sqrt 3 + 1 ))$

Applying property
$(a - b) (a + b) = a^{2} - b^{2}$ $color(blue)((a-b)(a+b) = a ^2 - b^2$ to the denominator we get:

$= \frac{5 \sqrt{3} + 5}{{(\sqrt{3})}^{2} - 1^{2}}$ $=(5sqrt 3 + 5 ) / ((sqrt3) ^2 -1^2)$

$= \frac{5 \sqrt{3} + 5}{3 - 1}$ $=(5sqrt 3 + 5 ) / (3 -1 )$

$= \frac{5 \sqrt{3} + 5}{2}$ $=(5sqrt 3 + 5 ) / 2$

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How do you rationalize the denominator and simplify $\frac{5}{\sqrt{3} - 1}$ $5/(sqrt3-1)$ ?

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