What is value of a^3 + (1/a^3) if (a + (1/a))^2 = 3?

[rahidz2003]

Ugh, this math problem is so annoying, I've tried solving for a and then plugging the result back, but the math gets complicated way too fast.

Here's the problem :

If \(\displaystyle (a+ (1/a))^2 = 3\), what is the value of \(\displaystyle a^3 + (1/(a^3))\)?

 

[masters]

Are you trying to solve: a +1/a[sup:13einznr]2[/sup:13einznr] = 3 ??

And then, whatever "a" turns out to be, plug it in the other expression.

 

[masters]

Or are you trying to solve: (a + 1/a)[sup:1i94h282]2[/sup:1i94h282] = 3 ??

 

[soroban]

Hello, rahidz2003!

If you're trying to solve for \(\displaystyle a\) first, no wonder you're having trouble.

This is one of those problems which has an elegant solution.

\(\displaystyle \text{If }\left(a+ \frac{1}{a}\right)^2 = \:3,\:\text{ find the value of }\,a^3 + \frac{1}{a^3}\)

\(\displaystyle \text{We have: }\;\left(a + \frac{1}{a}\right)^3 \:=\:3\quad\hdots\quad\text{Note that: }\:a + \frac{1}{a} \:=\:\sqrt[3]{3}\)


\(\displaystyle \text{Expand: }\;a^3 + 3a + \frac{3}{a} + \frac{1}{a^3} \:=\:3\)

. . \(\displaystyle \text{and we have: }\;a^3 + \frac{1}{a^3} +3\underbrace{\left(a+\frac{1}{a}\right)}_{\text{This is }\sqrt[3]{3}} \:=\:3\)

\(\displaystyle \text{Therefore: }\:a^3 + \frac{1}{a^3} + 3\sqrt[3]{3} \:\;=\;\:3\quad\Rightarrow\quad \boxed{a^3 + \frac{1}{a^3} \;\;=\;\;3 - 3\sqrt[3]{3}}\)