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Question
The value of \[\frac{\sqrt{48} + \sqrt{32}}{\sqrt{27} + \sqrt{18}}\] is
Options
\[\frac{4}{3}\]
4
3
`3/4`
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Solution
Given that `(sqrt48+sqrt32)/(sqrt27 +sqrt18)`
We know that rationalization factor for `sqrt27 +sqrt18` is`sqrt27 - sqrt18` We will multiply numerator and denominator of the given expression `(sqrt48+sqrt32)/(sqrt27 +sqrt18)` by`sqrt27 - sqrt18`, to get
`(sqrt48+sqrt32)/(sqrt27 +sqrt18) xx (sqrt27-sqrt18)/(sqrt27 -sqrt18) = (sqrt48 xx sqrt27 - sqrt48 xx sqrt18 + sqrt32 xx sqrt27 - sqrt32 xx sqrt18)/ ((sqrt27)^2 - (sqrt18)^2)`
We can factor irrational terms as
` (sqrt(3) xx sqrt16 xx sqrt9 xx sqrt3 - sqrt3 xx sqrt16 xx sqrt9 xx sqrt2 +sqrt2 xx sqrt16 xx sqrt3 xx sqrt9 - sqrt2 xx sqrt16 xx sqrt9 xx sqrt2)/((sqrt27)^2 - (sqrt18)^2)`
`= ((sqrt3)^2 xx 4 xx 3 - sqrt(3xx2)xx 4 xx 3 + sqrt(2 xx3) xx 4 xx3 - (sqrt2)^2 xx 4 xx 3)/(27-18) `
`= (3xx12-12xxsqrt6+12 xxsqrt6 -2 xx12)/(27-18) `
`= (36-12sqrt6+12sqrt6-24)/(27-18)`
`= 12/9`
` = 4/3`
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