Apply the laws of exponents to simplify the following:

3^3n / (27)^2n

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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

Therefore...

27^2n = (3^3)^2n

Powers of powers require exponents to be multiplied. The rule is:

(x^a)^b = x^ab

Therefore...

(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

X^A-X^B=X^(A-B)

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

(3^3)^2n=3^6n

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

3^(-3n)= (3^3)^-n=27^(-n)

by the law of negative exponents

27^(-n)=1/(27^n)

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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