Answer:
[tex]a^3 + 1 = 0[/tex]
Step-by-step explanation:
We start with the equation:
[tex]a + \frac{1}{a} = 1[/tex]
We want to find the value of:
[tex]a^3 + 1 =[/tex]
We can start with our previous equation and multiply both sides by a:
[tex](a + \frac{1}{a})*a = 1*a\\a^2 + 1 = a[/tex]
Now we can rewrite our initial expression as:
[tex]a = 1 - \frac{1}{a}[/tex]
Replacing that in the right side, we get:
[tex]a^2 + 1 = a = 1 - \frac{1}{a}[/tex]
Now again, let's multiply both sides by a
[tex]a*(a^2 + 1) = a*(1 - \frac{1}{a} )\\a^3 + a = a - a/a\\a^3 + a = a - 1\\a^3 = -1\\a^3 + 1 = 0[/tex]
So we can conclude that:
[tex]a^3 + 1 = 0[/tex]
