Solve r^3=8

$$r ^ { 3 } = 8$$

Solve for r

$$r=2$$

Solution Steps

Subtract 8 from both sides.

$$r^{3}-8=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -8 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±8,±4,±2,±1$$

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.

$$r=2$$

By Factor theorem, r-k is a factor of the polynomial for each root k. Divide r^{3}-8 by r-2 to get r^{2}+2r+4. Solve the equation where the result equals to 0.

$$r^{2}+2r+4=0$$

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, 2 for b, and 4 for c in the quadratic formula.

$$r=\frac{-2±\sqrt{2^{2}-4\times 1\times 4}}{2}$$

Do the calculations.

$$r=\frac{-2±\sqrt{-12}}{2}$$

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

$$r\in \emptyset$$

List all found solutions.

$$r=2$$