Hey..
Sorry for the horrible appearence of this question but here it goes..
Show that:
17(1-(1/17^2))^0.5 = n(sqrt(2))
Where n is an integer, whose value is to be found
the closest ive gotten so far is
17(sqrt(1-17^-2))
or
17(sqrt((17^2-1)/(17^2))
... but I'm not sure where to start since 17 is prime and I don't think the brackets can be expanded because the sqrt() surrounding the right-most part of the expression doesn't allow this as far as I'm aware
I'd prefer to have the steps to complete the problem since the value for n is given at the back of the textbook (n=12).
Thanks in advance,
Kwah =]
the trick is this: if x is positive, then . so if we square a positive number and then square root it, we simply undo the square root, so that makes sense. furthermore, it is a law of surds that:
so, ............by the first thing i said
.........by the law of surds that i mentioned
the rest follows
uhh... that is simply a less-clear explanation of the steps i did and naming it as the law of surds lol (dull)
btw, forgive me if ive misunderstood, but none of this quote from your post is accurate...
For , it does not matter if x is positive since it gets squared first...the trick is this: if x is positive, then .
Obviously, if it gets rooted first then squared, x has to be positive
As for this second sentence, it does not make sense that squaring a number THEN rooting it = undoing the root ((in fact, applying a root is undoing the square - which came first))so if we square a positive number and then square root it, we simply undo the square root, so that makes sense.
regards,
Kwah =]
it is a law of surds. and my explanation is more general than yours was. you used 17 as the specific example, my x could be anything.
you are mistaken. in general,btw, forgive me if ive misunderstood, but none of this quote from your post is accurate...
For , it does not matter if x is positive since it gets squared first...
Obviously, if it gets rooted first then squared, x has to be positive
by the way, x cannot be rooted first if it was negative. you'd get a complex number in that case
the square is under the square root, so we apply it first. , you do what is in the brackets first.As for this second sentence, it does not make sense that squaring a number THEN rooting it = undoing the root ((in fact, applying a root is undoing the square - which came first))
I'm really confused where to begin with replying to this ..
first off, i don't agree that
regardless of the value of , positive or negative, it results in , yeah?
roots can have a positive OR negative value, therefore does not necessarily equate to
If, as I stated in my last post, the is OUTSIDE the root, then is forced to become positive for two reasons..
is true (note: the outside the root)
for the reasons we have both stated, you cannot root negative numbers therefore x is required to be positive..
AND, squaring it afterwards would force it to become positive regardless..
the only other confusions / point I have to make about this is
(negative)
There was more that I wanted to say but it has gotten lost in my mind lol..
regards,
Kwah =]
by definition. it is not something i made up. there are two standard definitions for |x|, and that is one of them.
(exercise: try graphing and see what you get. it will be the absolute value function for x NOT the line y = x as you are claiming)
counter example to what you said.
let x = -2
compute
we get:
also, i never said anything about , so i do not see why you are arguing with me about it. yes indeed, the domain of this function is . that is if we are considering real numbers. i have no problem with you saying that x has to be nonnegative here for real numbers.
don't worry about itThere was more that I wanted to say but it has gotten lost in my mind lol..
try to remember and come back
jeez lol ..
is NOT correct ...
(im not sure how to get the plus minus symbol)
Think about it .. how did you get ??
err.. graphing would give the two lines y = x and y = -x(exercise: try graphing and see what you get. it will be the absolute value function for x NOT the line y = x as you are claiming)
I'm merely offering an alternative where it DOES have to equal |x|also, i never said anything about , so i do not see why you are arguing with me about it.
i thought i might respond to this specifically.
ALWAYS
it is never true that
i think i see where your misunderstanding comes from
for instance, if you are solving the solution is (remark here: how did i get +/- 1? if you take sqrt(x^2) to be |x| it makes sense! if we use say sqrt(x^2) = x, then the only solution would be x = 1, which we know is false)
however, that is not the same as saying which is in fact what you are claiming by saying the root of a positive number can be negative or positive.
do you understand what i am saying, or is there something else you're not clear on?
i never said -2^2 = 4, i said (-2)^2 = 4. that is correct
-2*-2 = +4
also, what you have is not correct. the square root function does not return negative values! look at the graph of square root x, the range is y greater than or equal to zero
no it will not. it will give the graph for |x|err.. graphing would give the two lines y = x and y = -x
you are mistaken my friendI'm merely offering an alternative where it DOES have to equal |x|
it is not me that is misunderstanding ..
use this numerical example ..
now root everything
///// EDIT: Just added this line in so that if can be directly compared to its algebraic equivalent contained in the "therefore"
therefore
or
EDIT: included the brackets to make it absolutely clear what I am intending here ..
and by "OR", i mean that either value is true - i do not know the plus OR minus symbol (a + sign with a - placed on top or below it)
Is this step-by-step proof clear enough?
regards
kwah =]
first of all so nothing you said after that is correct (in the sense that the proof started with an incorrect premise). you started with an incorrect assumption. if we take f(x) = x^2, then -x^2 = - f(x) not f(x)
it is true that which is completely different from what you are saying
it is funny though, you last statement agrees with me. |x| returns x or minus x depending on the sign of x
yeah .. that is what i meant .. for (-2) to be squared..
HAVING THE RANGE GREATER THAN OR EQUAL TO ZERO IS LIMITING TO NOT SHOW ANY VALUES LESS THAN ZERO WHICH IS WHY YOU ARE ONLY SEEING ONE VALUE, DUH!!also, what you have is not correct. the square root function does not return negative values! look at the graph of square root x, the range is y greater than or equal to zero