Division of x^n with y^n

Confusion does happen when you are bombarded with many numbers, exponential and its likes.

After dealing with indices, logarithm and their multiplication and division, the brain will sort of tangle up and produces weird happenings.

Take 2 examples below:

1) x^n / y^n ==> x /y

2) log x^n / log y^n ==> (log x) / (log y)

By looking at the first example, you may find nothing wrong.
Since the power "n" is similar for the numerator and denominator, you can do the normal cancellation as you do for "mx / my" = x/y.

However, something may tell you that something is amiss.

While "mx / my" is truely x / y, this is because mx means m times x.
There are "m" number of x that are ADDED up.

For x^n, it means "x" is multiplied by itself n times. (or x times x times x times x ....)

Thus x^n is not equal to xn.
The truth of "cancellation" is that since a / a = 1, and this "1" is not required to be written, the disappearance seems to be "cancellation".

Let me explain further with an example(A).
ax / ay = (a/a)(x/y) = (1) (x / y) = x / y. The "1" disappeared and seems to be cancelled off.

The mistake made in Example 1 in the beginning, is the assumption that the powers "n" followed the concept of "ax" in example(A).

Correct answer for x^n / y^n = (x/y)^n. ==> The powers of n are not removed.

Now for the Example 2, at the beginning, it showed a similar cancellation of the powers "n".
But this time round, it can be said to be conditionally correct.

Why?

If the idea that similar "letter" of "n" in the power can be removed through cancellation, then the answer, although correct, is theoretically wrong.

However, if you know that using the Power Law of logarithm, log x^n can become "nlogx" and therefore, log y^n can also be "nlog y", the result of (log x) /(log y) can be rightfully considered correct, since the "n" is removed according to the idea that n/n = 1 and disappeared, or qualified for removal.

In summary, mistakes do happen when the concept of power (x^n) and pure multiplication (x times n) is not clearly understood.

Cancellation of "letters" or symbols in math expressions should be highlighted as a shortcut to removal due to being "1" that can be omitted in the written form.

This concept of "cancellation" is easy if you understand that it is because of a/a = 1.

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