Rationalizing a denominator involving the sum of 3 cube roots

anemone

MHB POTW Director
Staff member
Hi members of the forum,

Problem:

Rationalize the denominator of $\displaystyle \frac{1}{a^\frac{1}{3}+b^{\frac{1}{3}}+c^{\frac{1}{3}}}.$

I know that if we are asked to rationalize, say, something like $\displaystyle \frac{1}{1+2^{\frac{1}{3}}}$, what we could do is the following:

$\displaystyle \frac{1}{1+2^{\frac{1}{3}}}=\frac{1}{1+2^{\frac{1}{3}}}.\frac{1-2^{\frac{1}{3}}+2^{\frac{2}{3}}}{1-2^{\frac{1}{3}}+2^{\frac{2}{3}}}$

$\displaystyle \frac{1}{1+2^{\frac{1}{3}}}=\frac{1-2^{\frac{1}{3}}+2^{\frac{2}{3}}}{1-2^{\frac{1}{3}}+2^{\frac{2}{3}}+2^{\frac{1}{3}}-2^{\frac{2}{3}}+2}$

$\displaystyle \frac{1}{1+2^{\frac{1}{3}}}=\frac{1-2^{\frac{1}{3}}+2^{\frac{1}{3}}}{1+2}$

$\displaystyle \frac{1}{1+2^{\frac{1}{3}}}=\frac{1-2^{\frac{1}{3}}+2^{\frac{1}{3}}}{3}$

But this one seems like not to be the case with the problem that I am asking on this thread...

Could anyone give me some hints to tackle it?

Fernando Revilla

Well-known member
MHB Math Helper
Hi members of the forum,Rationalize the denominator of $\displaystyle \frac{1}{a^\frac{1}{3}+b^{\frac{1}{3}}+c^{\frac{1}{3}}}.$
We can use the equality
$$(x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) = x^3 + y^3 + z^3 - 3xyz$$
If $x=\sqrt[3]{a},y=\sqrt[3]{b},z=\sqrt[3]{c}$ and we multiply numerator and denominator by $x^2 + y^2 + z^2 - xy - xz - yz$ we get in the denominator $a+b+c-3\sqrt[3]{abc}$. Now, is easier to rationalize.

anemone

MHB POTW Director
Staff member
We can use the equality
$$(x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) = x^3 + y^3 + z^3 - 3xyz$$
If $x=\sqrt[3]{a},y=\sqrt[3]{b},z=\sqrt[3]{c}$ and we multiply numerator and denominator by $x^2 + y^2 + z^2 - xy - xz - yz$ we get in the denominator $a+b+c-3\sqrt[3]{abc}$. Now, is easier to rationalize.
Thanks, Fernando Revilla! While I am so happy to receive your help in such a short period of time, I can see it now that what you suggested to me is basically using the same principle that I used to tackle the example that I cited above, it's such a shame of me for not thinking harder before posting...

Guess that when I get stuck, the only place that I could think of (instead of keep plugging away) is MHB!

MarkFL

Staff member
Many times the solution seems obvious...but only after it has been shown to us.

And your posting has given others useful information...I wouldn't have known how to rationalize that denominator without furrowing my brows and committing pen to paper...

Klaas van Aarsen

MHB Seeker
Staff member
Suppose you multiply numerator and denominator both with $a^3b^3c^3$.

MarkFL

Staff member
Suppose you multiply numerator and denominator both with $a^3b^3c^3$.
$\displaystyle a^{\frac{10}{3}}b^3c^3+a^3b^{\frac{10}{3}}c^3+a^3b^3c^{\frac{10}{3}}$
$\displaystyle a^{\frac{10}{3}}b^3c^3+a^3b^{\frac{10}{3}}c^3+a^3b^3c^{\frac{10}{3}}$