Division of powers require exponents to be subtracted. The rule is:
x^a / x^b = x^(a-b)
However, the bases of the powers must be equal. In this case, the base number is 3.
27 = 3^3
27^2n = (3^3)^2n
Powers of powers require exponents to be multiplied. The rule is:
(x^a)^b = x^ab
(3^3)^2n = 3^6n
So now we have the following fraction:
3^3n / 3^6n
Division of powers require subtraction of exponents. Therefore...
3^(3n - 6n) = 3^(-3n)
A negative exponent causes the power to become its reciprocal. Therefore...
3^(-3n) = 1 / (3^3n)
Simplified answer: 1 / (3^3n)
the laws of exponentials state that when SAME-based exponentials divide, the exponentials subtract each other. In math notation
well, in this case, we still do not see a common base. However, we know that 27 = 3^3
so the denominator (27)^2n= (3^3)^2n
by law of exponentials, a exponent's exponent result in the exponents timed together
the original equation becomes:
3^3n/3^6n which using the exponential law of division
= 3^(3n-6n)= 3 ^ (-3n)
well if your teacher allows you to write in negative exponents, ust keep the answer, if not, continue simplify:
by the law of mutiplying exponents again
by the law of negative exponents
Well, you could also change the numerator 3^3n into 27^n first so that the base is both 27, do whatever you think is easier.
Hope this helps
To benefit from properties of exponentials, we'll create matching bases to numerator and denominator.
Since the denominator is a multiple of numerator, we'll write 27 = 3^3
3^3n / (27)^2n = 3^3n/(3^3)^2n = 3^3n/3^(3*2n) = 3^3n/3^6n
Sincve the bases are matching, we'll perform the division, subtracting the exponents.
3^3n/3^6n = 3^(3n - 6n) = 3^(-3n)
We'll apply the negarive power property:
3^(-3n) = 1/3^3n = 1/27^n
Therefore, the simplified result is: 1/27^n