Algebra 2 - Math Brainbuster?
ok so one of my teachers told me there was a lie in our math text book. She told us anyone who finds a counter example will receive a couple of extra credit points in the class. My class has ended and for some odd reason she won't tell me the answer, so now i'm here.
my text book says:
if m is an integer and, n and k are natural numbers, and all indicated roots exist, then
k*n root a^k*m = n root a^m.
if this confusing "k*n root", think of when you have something to the cubed root.
cubed root 27 = 27^1/3 = 3.
best way i can think to explain it.
my book defines natural numbers as positive numbers not including 0
my teacher wants us to prove that the 1st equation does not equal the second equation.
Thanks in advance.
Update:@ N.D. Prabhakar
trust me i've asked tons of my friends to look at it most i find to be extremely intelligent and knowledgeable in math. All say the same thing you did, yet she wont prove to me how it is wrong.
Comments
If the equation is really
a^((km)/(kn)) = a^(m/n)
and k is nonzero, then there is no problem. By order of operations you simplify (km)/(kn) to m/n first, and then both sides are clearly the same.
But the way you wrote it the first time makes me think the equation should actually look like this:
(a^(km))^(1/kn) = (a^m)^(1/n)
Then there is a problem as long as you consistently pick the square root of 1 to be 1. For example, let a = -1, k = 2, m = n = 1.
Then
(a^(km)) = (-1)^2 = 1,
so (a^(km))^(1/kn) = 1^(1/2) = 1.
But (a^m)^(1/n) = ((-1)^1)^1 = -1
So the two sides are not equal.
i will walk you in the process the 1st, then you definately would desire to have no undertaking with something. Take one in each and every of your equations, and circulate a variable to a minimum of one area of the equals sign so this is by potential of itself. It purely works if the different equation has this variable additionally, like this: 5x + 4y = -9 Subtract 4y from the two factors to bypass it to the astonishing 5x = -9 -4y Lose the 5 by potential of dividing the two factors by potential of five 5x/5 = ( -9 - 4y) /5 5x/5 turns into basically x. x = ( -9 - 4y) /5 Now plug in what x is comparable to in the different undertaking like this. x - 3y = 21 and replace x (-9 - 4y)/5 - 3y = 21 Now remedy for y! (-9/5) - (4/5)y - 3y = 21 enable me understand in case you prefer help with this.
I am not sure what problem she finds in the statement.
As far as I can see, it is correct:
It is saying:
(a^(km))^(1/kn)
= a^(km/kn) using the rule that x^(M)^N=x^(M*N)
=a^(m/n) canceling k which is okay since k is not 0
a^(km)^(1/kn) is the kn-th root of a to the power km
a^(m/n) is the n-th root of a to the power m
I would be interested in seeing her counter-example. Perhaps you could post it, or send it to me. Thanks.
Ok, makes sense to me now. And I can't see the exception either. If you do ever find out what it is, I'd love to read it. I honestly don't see it, sorry I can't be any more help.