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

2 Answers
smendyka
May 18, 2018

See a solution process below:

Explanation:

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

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

(5 + sqrt(3))/(5 + sqrt(3)) * 2/(5 - sqrt(3)) =>

(2(5 + sqrt(3)))/((5 + sqrt(3)) * (5 - sqrt(3))) =>

(2 * 5 + 2sqrt(3))/((5 * 5) - 5sqrt(3) + 5 sqrt(3) - sqrt(3)sqrt(3)) =>

(10 + 2sqrt(3))/(25 - 0 - 3) =>

(10 + 2sqrt(3))/22 =>

(5 + sqrt(3))/11 =>

Or

5/11 + sqrt(3)/11

Sridhar V.
May 18, 2018

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

Explanation:

" "
Rationalize the denominator and simplify: color(blue)(2/(5-sqrt3)

"Given the rational expression: " color(red)(2/(5-sqrt3)

rArr 2/(5-sqrt3)*color(blue)([(5+sqrt3)/(5+sqrt3)]

rArr [(2*(5+sqrt(3)))/((5^2)-(sqrt(3))^2]]

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

In our problem: a=5 and b= sqrt(3)

rArr (2*(5+sqrt(3)))/(25-3)

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.

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