Basic Math Examples

$729 = s^{3}$

Step 1

Rewrite the equation as $s^{3} = 729$ .

$s^{3} = 729$

Step 2

Subtract $729$ from both sides of the equation.

$s^{3} - 729 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Rewrite $729$ as $9^{3}$ .

$s^{3} - 9^{3} = 0$

Step 3.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = s$ and $b = 9$ .

$(s - 9) (s^{2} + s \cdot 9 + 9^{2}) = 0$

Step 3.3

Simplify.

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Step 3.3.1

Move $9$ to the left of $s$ .

$(s - 9) (s^{2} + 9 s + 9^{2}) = 0$

Step 3.3.2

Raise $9$ to the power of $2$ .

$(s - 9) (s^{2} + 9 s + 81) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$s - 9 = 0$

$s^{2} + 9 s + 81 = 0$

Step 5

Set $s - 9$ equal to $0$ and solve for $s$ .

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Step 5.1

Set $s - 9$ equal to $0$ .

$s - 9 = 0$

Step 5.2

Add $9$ to both sides of the equation.

$s = 9$

Step 6

Set $s^{2} + 9 s + 81$ equal to $0$ and solve for $s$ .

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Step 6.1

Set $s^{2} + 9 s + 81$ equal to $0$ .

$s^{2} + 9 s + 81 = 0$

Step 6.2

Solve $s^{2} + 9 s + 81 = 0$ for $s$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.2.2

Substitute the values $a = 1$ , $b = 9$ , and $c = 81$ into the quadratic formula and solve for $s$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (1 \cdot 81)}}{2 \cdot 1}$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

Raise $9$ to the power of $2$ .

$s = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot 81}}{2 \cdot 1}$

Step 6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 81$ .

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Step 6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$s = \frac{- 9 \pm \sqrt{81 - 4 \cdot 81}}{2 \cdot 1}$

Step 6.2.3.1.2.2

Multiply $- 4$ by $81$ .

$s = \frac{- 9 \pm \sqrt{81 - 324}}{2 \cdot 1}$

Step 6.2.3.1.3

Subtract $324$ from $81$ .

$s = \frac{- 9 \pm \sqrt{- 243}}{2 \cdot 1}$

Step 6.2.3.1.4

Rewrite $- 243$ as $- 1 (243)$ .

$s = \frac{- 9 \pm \sqrt{- 1 \cdot 243}}{2 \cdot 1}$

Step 6.2.3.1.5

Rewrite $\sqrt{- 1 (243)}$ as $\sqrt{- 1} \cdot \sqrt{243}$ .

$s = \frac{- 9 \pm \sqrt{- 1} \cdot \sqrt{243}}{2 \cdot 1}$

Step 6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$s = \frac{- 9 \pm i \cdot \sqrt{243}}{2 \cdot 1}$

Step 6.2.3.1.7

Rewrite $243$ as $9^{2} \cdot 3$ .

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Step 6.2.3.1.7.1

Factor $81$ out of $243$ .

$s = \frac{- 9 \pm i \cdot \sqrt{81 (3)}}{2 \cdot 1}$

Step 6.2.3.1.7.2

Rewrite $81$ as $9^{2}$ .

$s = \frac{- 9 \pm i \cdot \sqrt{9^{2} \cdot 3}}{2 \cdot 1}$

Step 6.2.3.1.8

Pull terms out from under the radical.

$s = \frac{- 9 \pm i \cdot (9 \sqrt{3})}{2 \cdot 1}$

Step 6.2.3.1.9

Move $9$ to the left of $i$ .

$s = \frac{- 9 \pm 9 i \sqrt{3}}{2 \cdot 1}$

Step 6.2.3.2

Multiply $2$ by $1$ .

$s = \frac{- 9 \pm 9 i \sqrt{3}}{2}$

Step 6.2.4

The final answer is the combination of both solutions.

$s = - \frac{9 - 9 i \sqrt{3}}{2}, - \frac{9 + 9 i \sqrt{3}}{2}$

Step 7

The final solution is all the values that make $(s - 9) (s^{2} + 9 s + 81) = 0$ true.

$s = 9, - \frac{9 - 9 i \sqrt{3}}{2}, - \frac{9 + 9 i \sqrt{3}}{2}$