+9673 My cousin is challenging me in Mathematics. He gave me this question and I still can't solve it for 1 week.
He said that the solution is x = 8, I plugged x = 8 into the equation and it is correct. But I don't know how to get the answer. He told me the question and the answer and he told me to do the steps.
If x = 8, then substitute it into the equation and see what you get. This is what will get!
8^1/3 +(8 - 16)^1/3 =(8 - 8)^1/3
2 + (-8)^1/3 = 0^1/3
2 + 2(-1)^1/3=0 ????????
+259 \(x^{1/3} + (x-16)^{1/3} = (x-8)^{1/3}\)
Now take every side ^3
x + x - 16 = x - 8. I+16. I -x
x = 8
+259 Cant I take every side ^3? I know the answer looked really simple but who knows ^^
+356 (x)^1/3 + (x - 16)^1/3 = (x - 8)^1/3------------Cube each side
3 (x - 16)^(1/3) x^(2/3) + 2 x + 3 (x - 16)^(2/3) x^(1/3) - 16 = x - 8 -------------Isolate all radicals
3 (x - 16)^(1/3) x^(2/3) + 3 (x - 16)^(2/3) x^(1/3) = -x + 8
idk what to do from here LOL. Thats what we learned in school with square roots. But that is Algebra 2 and this is NOT algebra 2! I am about to go to school and gonna see if my teacher can do it and get back to you.
+118673 You know Ninja, you are a smart kid - yes I know you already know that LOL
I think you should start to learn some Latex. It is easy, you can learn it a little at a time and think how smart you will look then! :))
Take this line
(x)^1/3 + (x - 16)^1/3 = (x - 8)^1/3
It is not to the power of 1 it is to the power of 1/3 so put the 1/3 s in parentheses. Like this
(x)^{1/3} + (x - 16)^{1/3} = (x - 8)^{1/3}
Now copy that line onto your clipbard
now open the latex tab and copy it in.
like this
\((x)^{1/3} + (x - 16)^{1/3} = (x - 8)^{1/3}\)
Now if you copy what I have done you will have used LaTex for the first time :))
If you have questions make sure you ask.
We have had at least one boy on here before, I think he was a lot younger than you, and he got really good at LaTex.
+118673 Yes I could not do any LaTex until I was on this forum. I didn't learn the basics here because back then this forum did not have Latex. Mr Massow added it for us quite a bit later :)
But smart kids like you and Max and MathsGod can learn it really quickly if you put your mind to it. :)
+118673 I was looking at this expansion in Wolfram Alpha
expansion of (-a+x)^(1/3)
and the expansion is
There is an obvious pattern here except why are both the first two terms + and then the signs alternate.
Why is that ?
Perhaps I should have put this as a new question but it is sideways related to this question ://
+9673 To make the sign alternate:
\(\sqrt[3]{-a}=-\sqrt[3]{a}\)
At least this is true according to my knowledge.
+9673 Finally I have the answer!!!
\(x^{1/3}+(x-16)^{1/3}=(x-8)^{1/3}\\ x + x - 16 + 3 x^{1/3}(x-16)^{2/3}+3x^{2/3}(x-16)^{1/3}= x - 8\\ 3x^{1/3}(x-16)^{1/3}((x-16)^{1/3}+x^{1/3})=8-x\\ (x^{1/3}(x-16)^{1/3}(x-8)^{1/3})=\dfrac{8-x}{3}\\ x^3 - 24x^2 - 128x = \dfrac{(x-8)^3}{-27}\\ x^3 - 24x^2 - 128x = - \dfrac{x^3 + 24x^2 -64x + 512}{27}\\ 28x^3 - 672x^2 +(128\times 28-192)x - 512 = 0\\ 28x^3 - 672x^2 + 3648x - 512=0\\ (x-8)(7x^2 - 112x + 16)=0\\ x = 8\text{ or }8+12\sqrt{\dfrac{3}{7}}\text{ or }8-12\sqrt{\dfrac{3}{7}}\)
