Solve n^3=-216

$$n ^ { 3 } = - 216$$

Solve for n

$$n=-6$$

Solution Steps

Add 216 to both sides.

$$n^{3}+216=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 216 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±216,±108,±72,±54,±36,±27,±24,±18,±12,±9,±8,±6,±4,±3,±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$n=-6$$

By Factor theorem, n-k is a factor of the polynomial for each root k. Divide n^{3}+216 by n+6 to get n^{2}-6n+36. Solve the equation where the result equals to 0.

$$n^{2}-6n+36=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -6 for b, and 36 for c in the quadratic formula.

$$n=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 1\times 36}}{2}$$

Do the calculations.

$$n=\frac{6±\sqrt{-108}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$n\in \emptyset$$

List all found solutions.

$$n=-6$$