Question

Find the values of m and n if : 
`4^(2m) = ( root(3)(16))^(-6/n) = (sqrt8)^2`

Answer 1

`4^(2"m") = ( root(3)(16))^(-6/"n") = (sqrt8)^2`

⇒ `4^(2"m") = (sqrt8)^2`                    ....(1)

and

`(root(3)(16))^(-6/n) = (sqrt8)^2`     ....(2)

From (1)

`4^(2"m") = (sqrt8)^2`

⇒ `(2^2)^(2"m") = (sqrt(2^3))^2`

⇒ `2^(4"m") = [(2^3)^(1/2)]^2`

⇒ `2^(4"m") = [ 2^( 3 xx 1/2 )]^2`

⇒ `2^(4"m") =  2^( 3 xx 1/2 xx 2)`

⇒ `2^(4"m") = 2^3`

⇒ 4m = 3

⇒ m = `3/4`

From (2), We have

`(3sqrt(16))^(-6/"n") = (sqrt8)^2`

⇒ `( root(3)(2 xx 2 xx 2 xx 2))^(-6/"n") = (sqrt( 2 xx 2 xx 2))^2`

⇒ `( root(3)(2^4))^(-6/"n") = ( sqrt(2^3))^2`

⇒ `[(2^4)^(1/3)]^(-6/"n") = [(2^3)^(1/2)]^2`

⇒ `[2^(4/3)]^(-6/"n") = [2^(3/2)]^2`

⇒ `2^( 4/3 xx ( - 6/"n" ) = 2^(3/2 xx 2)`

⇒ `2^(-8/"n") = 2^3`

⇒ `-8/"n" = 3`

⇒ ` "n" = -8/3 "Thus m" = 3/4"n" = - 8/3`