How do you show that (3+sqrt2)/(5+sqrt8)3+25+8 can be written to (11-sqrt2)/1711217?

2 Answers
Aug 12, 2018

Please see below.

Explanation:

We know that ,

color(red)((1)a^2-b^2=(a-b)(a+b)(1)a2b2=(ab)(a+b)

Let ,

X=(3+sqrt2)/(5+sqrt8)X=3+25+8

Multiplying numerator and denominator by (5-sqrt8)(58)

X=(3+sqrt2)/(5+sqrt8)xx (5-sqrt8)/(5-sqrt8)X=3+25+8×5858

Simplifying we get

X=(3(5-sqrt8)+sqrt2(5-sqrt8))/((5)^2-(sqrt8)^2)tocolor(red)([because(1)])

:.X=(15-3sqrt8+5sqrt2-sqrt16)/(25-8)

:.X=(15-3*2sqrt2+5sqrt2- 4)/17to[becausesqrt8=sqrt(4xx2)=2sqrt2]

:.X=(11-6sqrt2+5sqrt2)/17

:.X=(11-sqrt2)/17

Aug 13, 2018

By Rationalization

Explanation:

(3 + sqrt2)/(5 + sqrt8)

By Rationalization, Note: a/(x + sqrty) = a/(x + sqrty) xx (x - sqrty)/(x - sqrty) or vice versa..

Hence;

(3 + sqrt2)/(5 + sqrt8) xx (5 - sqrt8)/(5 - sqrt8)

Rationalizing..

[(3 + sqrt2)(5 - sqrt8)]/[(5 + sqrt8)(5 - sqrt8)]

Expanding..

[(15 - 3sqrt8 + 5sqrt2 - sqrt16)]/[(25 - 8)]

Simplifying..

[(15 - 3sqrt8 + 5sqrt2 - 4)]/17

Further simplifying..

[(15 - 3sqrt8 + 5sqrt2 - 4)]/17

Collecting like terms..

[(15 - 4 - 3sqrt8 + 5sqrt2)]/17

Simplifying..

[(11 - 3sqrt(2 xx 4) + 5sqrt2)]/17

Further simplifying..

[(11 - 3(sqrt2 xx sqrt4)+ 5sqrt2)]/17

[(11 - 3(sqrt2 xx 2)+ 5sqrt2)]/17

[(11 - 3 xx 2(sqrt2)+ 5sqrt2)]/17

[(11 - 6sqrt2+ 5sqrt2)]/17

[(11 - sqrt2)]/17

Therefore;

(3 + sqrt2)/(5 + sqrt8) = (11 - sqrt2)/17

As required..