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

Question

Eva Adamson · Accepted Answer

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

Julia Adams · Answer

#1/2 (2sqrt6-5)# 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)#

Avery Alexander · Answer

undefined To simplify the expression (2√2 - 2√3)/(4√3 + 4√2), we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is (4√3 - 4√2).

Expanding the expression, we get:
[(2√2 - 2√3)(4√3 - 4√2)] / [(4√3 + 4√2)(4√3 - 4√2)]

Simplifying further, we have:
[8√6 - 8√2√3 - 8√6 + 8√2√3] / [16(3) - 16(2)]

Combining like terms, we get:
[-16√6] / [16]

Simplifying the expression, we have:
-√6 / 1

Therefore, the simplified form of (2√2 - 2√3)/(4√3 + 4√2) is -√6/16.