Question

If \[4x - 4 x^{- 1} = 24,\] then (2x)^x equals

Options

• \[5\sqrt{5}\]

• \[\sqrt{5}\]

• \[25\sqrt{5}\]

• 125

Answer 1

We have to find the value of  `(2x)^x`if  `4^x - 4^(x-1) = 24`

So,

Taking 4x as common factor we get 

`4^x (1- 1/4) = 24`

`4^x (1-4^-1) = 24`

`4^x ((1xx4)/(1 xx4)-1/4) = 24`

`4^4 ((4-1)/4)= 24`

`4^x xx 3/4 = 24`

`4^x = 24 xx 4/3`

`4^x = 32`

`2^(2x) =2^5`

By equating powers of exponents we get 

`2x = 5 `

`x=5/2`

By substituting `x=5/2`  in `(2x)^x` we get

`(2x)^x=(2xx 5/2)^(5/2)`

  = `(2xx5/2)^(5/2)`

`=5^(5/2)`

`=5^(5 xx1/2)`

`(2x)^x = 2sqrt(5^5)`

`=2sqrt (5xx5xx5xx5xx5)`

`= 5xx5  2sqrt5`

 = `25sqrt5`