Solving Polynomial: How to Expand Square Root

[jessawells]

hi,

can anyone show me how to solve this:

$$\sqrt{x^3} + \sqrt{1+x^3}$$

i want to get it to so that there's only 1 term of x. but i don't know how to expand the squared root. any help is appreciated.

 

[Jameson]

Usually when you say solve an equation, you mean something like solve x + 1 = 2.

Did you mean solve :$$\sqrt{x^3} = \sqrt{1+x^3}$$ (Probably not, since this is a false statement)

or simplify

$$\sqrt{x^3} + \sqrt{1+x^3}$$
?

I'm guessing the second one. Do you have an idea of how to approach it?

 

[jessawells]

hi,

yes, its the second case - simplifying the expression. I'm not sure at all how to approach it. I've thought about using the fact that both terms are squared - eg. making it $$\sqrt{x^3 + (1+x^3)}$$ but i know that's wrong. other than that, I've been trying to expand the $$\sqrt{1+x^3}$$ term, the way you would expand something like $$(1+x)^2$$, but I've had no success. any help would be great.

 

[Zurtex]

Well Mathematica can't simplify it.

 

[Jameson]

Agreed. That polynomial is in simplest form.

 

[Data]

it's not a polynomial, either.

 

[Jameson]

Thank you for giving us the correct definition of a polynomial.

 

[Data]

No... polynomials only have natural powers. A polynomial is something of the form

$$\alpha_0 + \alpha_1x + \alpha_2x^2 + \ . \ . \ . \ + \alpha_nx^n.$$

which $\sqrt{x^3} + \sqrt{x^3+1}$ certainly is not.

Now, you can certainly do some interesting things to $\sqrt{x^3} + \sqrt{x^3 + 1}$, as usual. For example,

$$\sqrt{x^3} + \sqrt{x^3 + 1} = \frac{1}{\sqrt{x^3+1}-\sqrt{x^3}}$$

$$= \sqrt{2\sqrt{x^3}\left(\sqrt{x^3}+\sqrt{1+x^3}\right) + 1}$$

but I would by no means consider those simpler.

 

[Hurkyl]

Please don't spam. You already posted your homework question here.