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\begin{document}
\title{Solving an Irrational Equation}
\author{Amit Yaron}
\date{June 10, 2021}

\maketitle
Wow, I see a sum of too cubic roots with square root inside. It looks
like a solution of a cubic equation using Cardano's formula. But under
the inner radical, there's ``x''. RHS is a constant. On YouTube
the length of the video clip is about 14 minutes. When there is a
number instead of ``x'', you can cube both sides and then form a
cubic equation solvable by searching for a rational root. Here we
do not need to solve a cubic equation. Let us solve it quickly with
no ads.

So, our equation is:

\[
\sqrt[3]{9+\sqrt{x}}+\sqrt[3]{9-\sqrt{x}}=3
\]

Solve by cubing both sides using the formula:

\[
(a+b)^{3}=a^{3}+b^{3}+3ab(a+b)
\]

If we take

\[
\begin{cases}
a= & \sqrt[3]{9+\sqrt{x}}\\
b= & \sqrt[3]{9-\sqrt{x}}
\end{cases}
\]

Then

\[
\begin{cases}
a^{3}= & 9+\sqrt{x}\\
b^{3}= & 9-\sqrt{x}\\
ab= & \sqrt[3]{(9-\sqrt{x})(9+\sqrt{x}}\\
a+b= & 3
\end{cases}
\]

Good! Now:

\begin{align*}
9+\sqrt{x}+9-\sqrt{x}+9\sqrt[3]{81-x} & =27\Rightarrow\\
\Rightarrow18+9\sqrt[3]{81-x} & =27\Rightarrow\\
\Rightarrow9\sqrt[3]{81-x} & =9\Rightarrow\\
\Rightarrow\sqrt[3]{81-x} & =1\Rightarrow\\
\Rightarrow81-x & =1\Rightarrow\\
\Rightarrow & x=80
\end{align*}

You can verify it using a calculator or a short expression in your
favorite scripting language's CLI. GNU Octave is a great one. Following
is the expression:
\begin{quotation}
cbrt(9+sqrt(80))+cbrt(9-sqrt(80))
\end{quotation}
The output is:
\begin{quotation}
ans = 3.0000
\end{quotation}

\end{document}