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Evaluate a^3 - \dfrac{1}{a^3} if a^2 - a - 1 = 0.

 Feb 20, 2024
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\({a}^{3}-\frac{1}{{a}^{3}}=(a-\frac{1}{a})({a}^{2}+1+\frac{1}{{a}^{2}})=(a-\frac{1}{a})({(a-\frac{1}{a})}^{2}+3)\), which is what we want to find.

We see \(a-\frac{1}{a}\) will me good to know.

We have:

\({a}^{2}-a-1=0\), divide both sides by a.

\(a-1-\frac{1}{a}=0\)\(a-\frac{1}{a}=1\).

Plugging in, \({a}^{3}-\frac{1}{{a}^{3}}=(1)({1}^{2}+3)=4\).

So, \(\underline{{a}^{3}-\frac{1}{{a}^{3}}}\) is 4.

 Feb 21, 2024

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