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Find the Values of M and N If : 4^(2m) = ( Root(3)(16))^(-6/N) = (Sqrt8)^2 - Mathematics

Sum

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

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Solution

`4^(2m) = ( root(3)(16))^(-6/n) = (sqrt8)^2`
⇒ `4^(2m) = (sqrt8)^2`                    ....(1)

and
`(root(3)(16))^(-6/n) = (sqrt8)^2`     ....(2)
From (1)
`4^(2m) = (sqrt8)^2`

⇒ `(2^2)^(2m) = (sqrt(2^3))^2`

⇒ `2^(4m) = [(2^3)^(1/2)]^2`

⇒ `2^(4m) = [ 2^( 3 xx 1/2 )]^2`

⇒ `2^(4m) =  2^( 3 xx 1/2 xx 2)`

⇒ `2^(4m) = 2^3`

⇒ 4m = 3

⇒ m = `3/4`

From (2), We have
`(3sqrt(16))^(-6/x) = (sqrt8)^2`

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

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

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

⇒ `[2^(4/3)]^(-6/x) = [2^(3/2)]^2`

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

⇒ `2^(-8/x) = 2^3`

⇒ `-8/x = 3`

⇒ ` x = -8/3 "Thus m" = 3/4x = - 8/3`

Concept: Solving Exponential Equations
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APPEARS IN

Selina Concise Mathematics Class 9 ICSE
Chapter 7 Indices (Exponents)
Exercise 7 (B) | Q 5 | Page 100
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