What is the cube root of 55555?

$38^3 = 54872$ and $39^3 = 59319$

$55555$ lies between them, so $\sqrt[3]{55555}$ is between $38$ and $39$.

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Quick estimate: $\sqrt[3]{N} \approx a + \dfrac{N - a^3}{3a^2}$, with $a = 38$:

$\sqrt[3]{55555} \approx 38 + \dfrac{55555 - 54872}{3 \times 38^2}$

$= 38 + \dfrac{683}{4332}$

$\approx 38 + 0.1577$

$= 38.1577$

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This estimate is very slightly above the true value (by less than $0.001$), so:

$\sqrt[3]{55555} = 38.157\ldots$

Rounded to two decimal places, $\sqrt[3]{55555} = 38.16$.