show :2 to the power of n =phi to the power of n=K to the power of n?

Answer 1

I'm PRESUMING that you meant (((2^Phi)^K)^n) = 2^(Phi*K*n) where * means "multiplied by the next number." [### But see my postscript for an alternative and very different interpretation.]

In other words, the result is "2 to the power (Phi*K*n), the PRODUCT of all the exponents." [Here, " * " means "multiply," a fairly commonly used symbol in computer text.]

For example, let Phi = 2, K = 3, and n = 4.

Then, working from the inside of the whole expression outwards, so that one uses the successive powers in the order Phi, then K, and finally n:

2^Phi = 2^2 = 4; 4^K = 4^3 = 64; and 64^4 = 16777216. (This last number needed a calculator!)

Let's now check that this is indeed 2^(Phi*K*n).

To do that we'll need (Phi*K*n), which is 2x3x4 = 24. O.K., so what is 2^24?:

According to my calculator, 2^24 = 16777216. It checks out!

So there you have it. Once you've raised a number to some initially given power, all further successive raisings of that result to yet more powers simply involve multiplying all of the successive power exponents together.

Live long and prosper.

### By the way, the symbol usually employed for "raised to the power p" on this service or on computer math lay-up software like Latek, etc., is the "carat", or " ^ ". You can read it as saying "up there!" It means any superscript, as typesetters would also describe it. The corresponding symbol for a subscript is the "underline" symbol,
" _ ". Thus "log_a (xyz^2)" would mean "log to the base 'a' of (x times y times [z squared])."

Note also how parentheses or brackets are used where they are needed to avoid confusion and/or ambiguity.

I hope that I interpreted what you were really asking about, in your question, correctly. However, there is the possiblity of lots of ambiguous (mis?) readings, from the way that you set it.

For example, without any of the needed delimiters, you COULD have been enquiring about 2^(phi^[K^n]). THAT final power, phi^[K^n] = 2.41785... x 10^24 ! Then 2^(phi^[K^n]) = 2^[2.41785... x 10^24]. This is an INCREDIBLY HUGE NUMBER, VERY MUCH LARGER than 16777216. In fact, it has over 10 MILLION digits BEFORE the decimal point --- I don't think that Yahoo! would like it if I tried to type it! (In fact, my calculator couldn't handle it.)

I hope you now see how important it is to write your mathematical expressions clearly, so that they can be properly understood. And even if you're forced to use words, still try to properly delimit expressions to avoid needless ambiguity.

As with any writing done for others to read, the goals to keep in mind are "Will intended readers understand me? Am I making myself as clear as I can?" That's why editing exists !

Answer 2

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Answer 3

(((2**Phi)**K)**n), where ** = power.
or
(((2^Phi)^K)^n), where ^ = power