what is the concept around an exponent or root that is a decimal.
its cool for example 125^2 is telling me "times me by myself"
125^3 is telling me "times me by myself and do it again!"
but what is 125^2.5 telling me to do?
what is the concept around an exponent or root that is a decimal.
its cool for example 125^2 is telling me "times me by myself"
125^3 is telling me "times me by myself and do it again!"
but what is 125^2.5 telling me to do?
Some of this is easy, some of it isn't.
Decimals that can be readily turned into fractions have an immediate interpretation. For example, you can show that
$\displaystyle a^{1/n} = \sqrt[n]{a}$
So your example
$\displaystyle 125^{2.5} = 125^{5/2} = \sqrt{125^5}$
If the decimal is something like $\displaystyle \sqrt{2}$ (ie. irrational) I am much more hazy about the interpretation. We can, however, do this:
$\displaystyle a^{\sqrt{2}} = a^{1.41421...}$
$\displaystyle = a^{1 + 0.4 + 0.01 + 0.004 + 0.0002 + 0.00001 + ...}$
$\displaystyle = (a^1)(a^{4/10})(a^{1/100})(a^{4/1000})(a^{2/10000})(a^{1/100000})...$
But that's still pretty messy to work with.
-Dan
fractional exponents get a little weird.
Note that $\displaystyle 125^{2.5} = 125^{5/2}$
now what this says is "times me by myself and do it again and again and again, and while you at it, take my square root (but that's if you interpret me as $\displaystyle \left( 125^5\right)^{1/2}$). If you interpret me as $\displaystyle \left( 125^{1/2} \right)^5$, then take my square root and then raise me to the 5th power"
thus, $\displaystyle 125^{2.5} = 125^{5/2} = \sqrt{125^5} = \left( \sqrt{125} \right)^5$
if the power is irrational (cannot be expressed as a fraction, then, you need your calculator)
see post #3 here for more info
EDIT: Thanks for letting me waste my time answering this, Dan
hehe, you know you caused him to pass you
i will try, this thing stops me from adding rep sometimes. i've been trying to add rep to Krizalid forever (i just tried to add rep to you and it wouldn't allow me to)
TPH went offline, hopefully next time he logs on he remembers how many points he had before i added rep
definitely rack your brain. the osmosis takes care of itself. you'd find that sometimes after racking your brain for a long time, you will give up and go to something else, when suddenly, BAM! with no warning, it comes to you. (your subconscious will get the message that figuring this out is important to you because you were racking your brain, and so it works on it while you're not noticing)
you should know concepts like these before going to calculus, you do not want to be in calculus if you do not have basic proficiency in algebra.would i be in better position to sink these concepts after i learned calculus?
got it:
125^2.5 says
split 125 into two equal parts that when multiplied equals 125:
$\displaystyle 11.2*11.2=125$
take one of those parts (11.2) and multiply that by 125 and 125 again.
likewise 125^2.75 says
split 125 into four parts that when multiplied together equals 125
$\displaystyle 3.3*3.3*3.3*3.3=125$
take three of those parts $\displaystyle 3.3*3.3*3.3=37.4$ and multiply by 125 and 125 again.
im going to watch cartoons.
I made a calculator to determine your reputation (I assume the same weights are default for all vbulletin sites, and that neither this site's weight nor the site I wrote it for's weight was modified. If this is not the case, then the calculator will tell you the wrong answer)
calculate your power of reputation - Starcraft Dream
The exact specifications for determining can be found here, but you need admin CP access to find the numbers vBulletin Manual
Or, if you plug your numbers into the equation, and they come out to roughly 50 points, then it is probably accurate.
Do you mean how to calculate it?
Probably the simplest thing to do is to get the decimal representation of $\displaystyle \sqrt{2} = 1.41421...$.
Thus
$\displaystyle 5^{\sqrt{2}} = 5^{1.41421...} = (5^1)(5^{0.4})(5^{0.01})(5^{0.004})(5^{0.0002})(5^ {0.00001})...$
$\displaystyle = 5 \cdot 5^{4/10} \cdot 5^{1/100} \cdot 5^{4/1000} \cdot 5^{2/10000} \cdot 5^{1/100000} \cdot ~...$
So you can calculate the first several terms in the product and get at least a decent approximation to it. (Note as the series progresses to smaller and smaller fractions $\displaystyle 5^n$ approaches 1.) But you can't get an exact value for it.
Plugging it into my calculator gives $\displaystyle 5^{\sqrt{2}} \approx 9.73852$
-Dan