pls solve this question fast.solve for x
Q. ${\left(x+27\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}+{\left(55-x\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}=4$

Dear Student,

Please find below the solution to the asked query:

We have : ${\left(x + 27 \right)}^{\frac{1}{4}} + {\left(55 - x \right)}^{\frac{1}{4}} = 4$

Here we can see that we get ' 4 '  by sum of two terms , So we assume possible combinations of integers that add upto give ' 4 '

i ) 4  and 0  , As :  4 + 0 = 4 

But we know :  4⁴ =  256  , So we can see that the same number 'x'  we add into ' 27 '  and subtract from ' 55 ' will never gives 256 , So we neglect that combination .

ii ) 2 and 2  , As : 2 + 2 = 4

We know :  2⁴ = 16 , So we can see that the same number 'x'  we add into ' 27 '  and subtract from ' 55 ' will never gives 16 , So we neglect that combination .

iii ) 1 and 3 , As :  1 + 3 = 4

We know : 3⁴ = 81  and 1⁴ = 1

Then ,

27 + x  = 1 , 

x  =  - 26

And

55 - x  = 81

x  = - 26

We can see that both terms gives a same number ' - 26 ' , And we can verify that ( At x  = - 26 ) , As :

$$\begin{gathered} {\left(- 26 + 27 \right)}^{\frac{1}{4}} + {\left(55 - \left( - 26\right) \right)}^{\frac{1}{4}} = 4 \\ \Rightarrow {\left(1 \right)}^{\frac{1}{4}} + {\left(55 + 26 \right)}^{\frac{1}{4}} = 4 \\ \Rightarrow 1 + {\left(81 \right)}^{\frac{1}{4}} = 4 \\ \Rightarrow 1+ {\left(3^4\right)}^{\frac{1}{4}} = 4 \\ \Rightarrow 1+3= 4 ( \mathrm{We} \mathrm{know} : {\left({\mathrm{a}}^{\mathrm{m}}\right)}^{\mathrm{n}} = {\mathrm{a}}^{\mathrm{m}\times \mathrm{n}} ) \\ \mathbf{\Rightarrow }\mathbf{4}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{4}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Verified}\mathbf{ }\mathbf{)} \end{gathered}$$

iv) 3 and 1 , As :  3 + 1 = 4

We know : 3⁴ = 81  and 1⁴ = 1

Then ,

27 + x  = 81 , 

x  = 54

And

55 - x  = 1

x  = 54

We can see that both terms gives a same number ' 54 ' , And we can verify that ( At x  = 54 ) , As :

$$\begin{gathered} {\left(54 + 27 \right)}^{\frac{1}{4}} + {\left(55 - 54 \right)}^{\frac{1}{4}} = 4 \\ \Rightarrow {\left(81 \right)}^{\frac{1}{4}} +{\left(1\right)}^{\frac{1}{4}} = 4 \\ \Rightarrow 3 + 1 = 4 \\ \mathbf{\Rightarrow }\mathbf{4}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{4}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Verified}\mathbf{ }\mathbf{)} \end{gathered}$$

Therefore,

x  = - 26  and  54                                                                   ( Ans )

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