Find the value of
See The picture
Answers
Step-by-step explanation:
Given:-
[(9^n ×3^2×3^n-27^n)]/[(3^3)^5×2^3] = 1/27
To find:-
Find the value of n ?
Solution:-
Given equation is
[(9^n ×3^2×3^n-27^n)]/[(3^3)^5×2^3] = 1/27
=> [(9^n×3^n×3^²-27^n)]/[(3^3)^5×2^3] = 1/27
=>[(9×3)^n ×3^2-27^n)]/[(3^3)^5×2^3] = 1/27
Since (ab)^m = a^m × b^m
=> [(27^n ×3^2-27^n)]/[(3^3)^5×2^3] = 1/27
=>[(27^n)( 3^2-1)]/[(3^3)^5×2^3] = 1/27
=>[(27^n) (9-1)]/[(3^3)^5×2^3] = 1/27
=>[(27^n) (8)]/[(3^3)^5×2^3] = 1/27
=>[(27^n) (2^3)]/[(3^3)^5×2^3] = 1/27
On cancelling 2^3 in both the numerator and the denominator
=> (27^n)/[(3^3)^5] = 1/27
=> (27^n)/((3^(3×5)) = 1/27
Since (a^m)^n = a^(mn)
=> (27^n) / (3^15 ) = 1/27
On applying cross multiplication then
=> 27^n ×27 = 3^15 × 1
=>27^(n+1) = 3^15
Since a^m × a^n = a^(m+n)
=> (3^3)^(n+1) = 3^15
=> 3^{3(n+1)} = 3^15
Since (a^m)^n = a^(mn)
=> 3^(3n+3) = 3^15
=> 3n+3 = 15
Since the bases are equal then exponents must be equal.
=> 3n = 15-3
=> 3n = 12
=> n = 12/3
=> n = 4
Therefore, n = 4
Answer:-
The value of n for the given problem is 4
Check:-
If n = 4 then ,
LHS = [(9^n ×3^2×3^n-27^n)]/[(3^3)^5×2^3]
=> [(9^4 ×3^2×3^4-27^4)]/[(3^3)^5×2^3]
=> [(27^4 ×3^2-27^4)]/[(3^3)^5×2^3]
=> [(27^4) ×(3^2-1)]/[(3^3)^5×2^3]
=>[(27^4) ×(8)]/[(3^3)^5×2^3]
=>(27^4) / (3^3)^5
=> (27)^4/(27)^5
=> (27)^(4-5)
=> 27^(-1)
=> 1/27
Since a^-n = 1/a^n
=> RHS
LHS = RHS is true for n = 4
Verified the given relations in the given problem
Used formulae:-
- a^m × a^n = a^(m+n)
- a^m / a^n = a^(m-n)
- (a^m)^n = a^(mn)
- (ab)^m = a^m × b^m
- a^-n = 1/a^n
- If the bases are equal then exponents must be equal.
Answer:
hope it may help you
have a nice day
