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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)
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
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
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