Question

If (4/9)^(m+n) = (2/3)^-6

What is the value of m & n if m = 2n​

Answer 1

Answer:

Option (b)

Step-by-step explanation:

Given :-

(4/9)^(m+n) = (2/3)^-6

To find :-

What is the values of m & n if m = 2n ?

Solution :-

Given that

(4/9)^(m+n) = (2/3)^-6

=> [(2/3)^2]^(m+n) = (2/3)^-6

We know that

(a^m)^n = a^(mn)

=> (2/3)^{2(m+n)} = (2/3)^-6

We know that

If the bases are equal then exponents must be equal.

=> 2(m+n) = -6

=> m+n = -6/2

=> m+n = -3 --------(1)

Given that m = 2n

If m = 2n then (1) becomes

=> 2n+n = -3

=> 3n = -3

=> n = -3/3

=> n = -1

Therefore, n = -1

If n = -1 then 2n = 2(-1) = -2

Therefore, m = -2

Therefore, m = -2 and n = -1

Answer:-

The values of m and n are -2 and -1 respectively.

Check:-

If m = -2 and n = -1 then LHS of the given equation

=> (4/9)^(-2-1)

=> (4/9)^-3

=> [(2/3)^2]^-3

=> (2/3)^-6

=> RHS

=> LHS = RHS is true for m = -2 and n = -1

and

m = -2 = 2(-1) = 2n

=> m = 2n

Verified the given relations in the given problem.

Used formulae:-

→ (a^m)^n = a^(mn)

→ If the bases are equal then exponents must be equal. i.e. If a^m = a^n => m = n