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How do you rationalize the denominator and simplify $\frac{1}{\sqrt{3} - 2}$ $1/(sqrt3-2)$ ?

1 Answer

Sep 30, 2015

Multiply the numerator and denominator by bye conjugate of the denominator to get:
$XXX - (\sqrt{3} + 2)$ $color(white)("XXX")-(sqrt(3)+2)$

Explanation:

The conjugate of a two-term expression $(a + b)$ $(a+b)$ is $(a - b)$ $(a-b)$ and visa versa.
The product of conjugates $(a + b) \times (a - b)$ $(a+b)xx(a-b)$ is $a^{2} - b^{2}$ $a^2-b^2$

For the given example, the conjugate of $(\sqrt{3} - 2)$ $(sqrt(3)-2)$ is $(\sqrt{3} + 2)$ $(sqrt(3)+2)$

$\frac{1}{\sqrt{3} - 2}$ $1/(sqrt(3)-2)$
$XXX = \frac{1}{\sqrt{3} - 2} \times \frac{\sqrt{3} + 2}{\sqrt{3} + 2}$ $color(white)("XXX")=1/(sqrt(3)-2)xx(sqrt(3)+2)/(sqrt(3)+2)$

$XXX = \frac{\sqrt{3} + 2}{({\sqrt{3}}^{2} - 2^{2})}$ $color(white)("XXX")=(sqrt(3)+2)/((sqrt(3)^2-2^2)$

$XXX = \frac{\sqrt{3} + 2}{3 - 4}$ $color(white)("XXX")=(sqrt(3)+2)/(3-4)$

$XXX = \frac{\sqrt{3} + 2}{- 1}$ $color(white)("XXX")=(sqrt(3)+2)/(-1)$

$XXX = - (\sqrt{3} + 2)$ $color(white)("XXX")=-(sqrt(3)+2)$

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