Question

How do you simplify `(2sqrt2- 2sqrt3 )/(4sqrt3+4sqrt2)`?

Answer 1

`"The answer is:" \qquad \qquad \qquad \quad { 2 sqrt{2} - 2 sqrt{3} }/{ 4 sqrt{3} - 4 sqrt{2} } \ = \ - { 1 }/{ 2 }.`

Explanation:

`"This one goes quite nicely, thankfully. A minor adjustment of"`
`"the given can add a bit of computational complexity. Anyway,"`
`"here we go:"`

`\qquad \qquad \qquad { 2 sqrt{2} - 2 sqrt{3} }/{ 4 sqrt{3} - 4 sqrt{2} } \ = \ { 2 ( sqrt{2} - sqrt{3} ) }/{ 4 ( sqrt{3} - sqrt{2} ) }`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ { 2 (-1)( sqrt{3} - sqrt{2} ) }/{ 4 ( sqrt{3} - sqrt{2} ) }`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ { 2 (-1) color{red}cancel{ ( sqrt{3} - sqrt{2} ) } }/{ 4 color{red}cancel{ ( sqrt{3} - sqrt{2} ) } }`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ { 2 (-1) }/{ 4 }`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad = \ - { 1 }/{ 2 }.`

`"This is our answer !!"`

`"So, we have found:"`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad { 2 sqrt{2} - 2 sqrt{3} }/{ 4 sqrt{3} - 4 sqrt{2} } \ = \ - { 1 }/{ 2 }.`

Answer 2

`1/2 (2sqrt6-5)`

Explanation:

Given, `[2 sqrt2 - 2 sqrt3]/[4 sqrt3 +4 sqrt2]`
`rArr [2(sqrt 2-sqrt 3)]/[4(sqrt3+sqrt2)]`
`rArr 1/2 (sqrt2-sqrt3)/(sqrt3+sqrt2)`
`rArr 1/2 [(sqrt2-sqrt3)(sqrt2-sqrt3)]/[(sqrt3+sqrt2)(sqrt2-sqrt3)]`
`rArr 1/2[(sqrt2)^2-2.sqrt2.sqrt3+(sqrt3)^2]/[2-3+sqrt6-sqrt6]`
`rArr 1/2 [5-2sqrt6]/(-1)`
`rArr 1/2(2sqrt6-5)`