If (9^n×3^2×(3^-n/2)-(27)^n)/(3^m×2^3)=1/27, Prove that m-n=1.
I suspect typos
The numerator includes a √3
the denominator is 8*3^m, which includes a power of 2
There is no way the result can be 1/27
If n=1,
(9^n×3^2×(3^-n/2)-(27)^n)
= (9^1×3^2×(3^-1/2)-(27)^1)
= (9*9/√3 - 27)
= 27(√3-1)
see where you messed up
The question is to prove m-n=1.
I myself feel there is typo in the denominator it should 3^3m and not 3^m rest everything is okay. I tried solving i got as m-3n=3.
To prove that m - n = 1 using the given equation, we need to simplify the equation and examine the exponents of the variables.
Let's break down the equation step by step:
(9^n × 3^2 × (3^(-n/2)) - (27)^n) / (3^m × 2^3) = 1/27
To simplify this equation, let's separately simplify the numerator and denominator before combining them.
Numerator:
9^n × 3^2 × (3^(-n/2)) - (27)^n
First, we can rewrite 9 as (3^2) and 27 as (3^3):
(3^2)^n × 3^2 × (3^(-n/2)) - (3^3)^n
Applying the exponent rule of (a^m)^n = a^(m × n), we get:
3^(2n) × 3^2 × (3^(-n/2)) - 3^(3n)
Next, using the exponent rule of a^m × a^n = a^(m + n), we can combine the exponents:
3^(2n + 2) × 3^(-n/2) - 3^(3n)
Now, let's simplify further using the exponent rule of a^(-n) = 1 / (a^n):
3^(2n + 2) × 1 / (3^(n/2)) - 3^(3n)
Applying the rule of a^m / a^n = a^(m - n), we can rewrite the division:
(3^(2n + 2)) × (3^(-n/2)) - 3^(3n) / (3^(n/2))
Using the rule of a^m × a^n = a^(m + n) for the first part:
3^(2n + 2 - n/2) - 3^(3n) / (3^(n/2))
Let's simplify the denominator:
Denominator:
3^m × 2^3
Applying the exponent rule of a^n × b^n = (a × b)^n:
(3 × 2)^m × 2^3
Simplifying further:
6^m × 2^3
Now, let's substitute the simplified numerator and denominator back into the equation:
(3^(2n + 2 - n/2) - 3^(3n) / (3^(n/2))) / (6^m × 2^3) = 1/27
Let's simplify the right side of the equation:
1/27 = (6^(-3)) × (2^(-3))
Now, let's compare the left and right sides of the equation:
(3^(2n + 2 - n/2) - 3^(3n) / (3^(n/2))) / (6^m × 2^3) = (6^(-3)) × (2^(-3))
To compare the exponents, we need to set them equal to each other:
2n + 2 - n/2 = -3 (1)
m = -3 (2)
Comparing the exponent of 3:
n - n/2 = -3/2 (3)
Let's simplify equation (3):
Multiplying through by 2:
2n - n = -3
n = -3
Substituting equation (2) into equation (1):
-3 + 2 = -3
-1 = -3
Since -1 does not equal -3, we can conclude that m - n does not equal 1. Therefore, the given equation (9^n × 3^2 × (3^(-n/2)) - (27)^n) / (3^m × 2^3) = 1/27 does not prove that m - n = 1.