Question

Let r, s, and t be the real roots of
(x - \sqrt[3]{13})(x - \sqrt[3]{53})(x - \sqrt[3]{103}) = 0.
Compute r^3 + s^3 + t^3.

Answer 1

To find the value of \( r^3 + s^3 + t^3 \) where \( r, s, t \) are the real roots of the equation

\[
(x - \sqrt[3]{13})(x - \sqrt[3]{53})(x - \sqrt[3]{103}) = 0,
\]

we first identify the roots. The roots are given directly as:

\[
r = \sqrt[3]{13}, \quad s = \sqrt[3]{53}, \quad t = \sqrt[3]{103}.
\]

We can compute \( r^3, s^3, \) and \( t^3 \) as follows:

\[
r^3 = 13, \quad s^3 = 53, \quad t^3 = 103.
\]

Now we calculate \( r^3 + s^3 + t^3 \):

\[
r^3 + s^3 + t^3 = 13 + 53 + 103.
\]

We perform the addition:

\[
13 + 53 = 66,
\]
\[
66 + 103 = 169.
\]

Thus, we have:

\[
r^3 + s^3 + t^3 = 169.
\]

Therefore, the value of \( r^3 + s^3 + t^3 \) is

\[
\boxed{169}.
\]