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Question
If \[4x - 4 x^{- 1} = 24,\] then (2x)x equals
Options
\[5\sqrt{5}\]
\[\sqrt{5}\]
\[25\sqrt{5}\]
125
Solution
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`
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