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

Question

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

2 Answers

Impact of this question

4622 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-05-18T13:37:50

See a solution process below:

Explanation:

To rationalize the denominator we must multiply the fraction by the appropriate form of $1$ $1$ to eliminate the radicals from the denominator.

For a denominator of the type $(a - b)$ $(a color(red)(-) b)$ we need to multiply by $(a + b)$ $(a color(red)(+) b)$ :

$\frac{5 + \sqrt{3}}{5 + \sqrt{3}} \cdot \frac{2}{5 - \sqrt{3}} \Rightarrow$ $(5 + sqrt(3))/(5 + sqrt(3)) * 2/(5 - sqrt(3)) =>$

$\frac{2 (5 + \sqrt{3})}{(5 + \sqrt{3}) \cdot (5 - \sqrt{3})} \Rightarrow$ $(2(5 + sqrt(3)))/((5 + sqrt(3)) * (5 - sqrt(3))) =>$

$\frac{2 \cdot 5 + 2 \sqrt{3}}{(5 \cdot 5) - 5 \sqrt{3} + 5 \sqrt{3} - \sqrt{3} \sqrt{3}} \Rightarrow$ $(2 * 5 + 2sqrt(3))/((5 * 5) - 5sqrt(3) + 5 sqrt(3) - sqrt(3)sqrt(3)) =>$

$\frac{10 + 2 \sqrt{3}}{25 - 0 - 3} \Rightarrow$ $(10 + 2sqrt(3))/(25 - 0 - 3) =>$

$\frac{10 + 2 \sqrt{3}}{22} \Rightarrow$ $(10 + 2sqrt(3))/22 =>$

$\frac{5 + \sqrt{3}}{11} \Rightarrow$ $(5 + sqrt(3))/11 =>$

Or

$\frac{5}{11} + \frac{\sqrt{3}}{11}$ $5/11 + sqrt(3)/11$

Answer 2 · 2018-05-18T13:51:15

$" "$
$\frac{2}{5 - \sqrt{3}} = \frac{5 + \sqrt{3}}{11}$ $color(brown)(2/(5-sqrt3) = [5+sqrt(3)]/11$

Explanation:

$" "$
Rationalize the denominator and simplify: $\frac{2}{5 - \sqrt{3}}$ $color(blue)(2/(5-sqrt3)$

$Given the rational expression: \frac{2}{5 - \sqrt{3}}$ $"Given the rational expression: " color(red)(2/(5-sqrt3)$

$\Rightarrow \frac{2}{5 - \sqrt{3}} \cdot [\frac{5 + \sqrt{3}}{5 + \sqrt{3}}]$ $rArr 2/(5-sqrt3)*color(blue)([(5+sqrt3)/(5+sqrt3)]$

$\Rightarrow ⎡ ⎢ ⎢ ⎣ \frac{2 \cdot (5 + \sqrt{3})}{(5^{2}) - {(\sqrt{3})}^{2}} ⎤ ⎥ ⎥ ⎦$ $rArr [(2*(5+sqrt(3)))/((5^2)-(sqrt(3))^2]]$

**Identity used: ** : $(a + b) (a - b) = a^{2} - b^{2}$ $color(green)((a+b)(a-b)=a^2-b^2$

In our problem: $a = 5 and b = \sqrt{3}$ $a=5 and b= sqrt(3)$

$\Rightarrow \frac{2 \cdot (5 + \sqrt{3})}{25 - 3}$ $rArr (2*(5+sqrt(3)))/(25-3)$

$\Rightarrow \frac{2 (5 + \sqrt{3})}{22}$ $rArr (2(5+sqrt(3)))/(22)$

$rArr (cancel 2(5+sqrt(3)))/(cancel 22^color(blue)(11))$

$rArr (5+sqrt(3))/11$

Hence,

$color(brown)(2/(5-sqrt3) = [5+sqrt(3)]/11$

Hope it helps.

Science

Math

Humanities

... and beyond

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

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you rationalize the denominator and simplify 2/(5-sqrt3)25−√32/(5-sqrt3)?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

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