Apply the laws of exponents to simplify the following: 3^3n / (27)^2nShow complete solution to explain the answer.

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samhouston's profile pic

Posted on

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)

shaznl1's profile pic

Posted on

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

 

giorgiana1976's profile pic

Posted on

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

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