Algebra Examples

$9^{6} = x^{3} \cdot 9^{3}$

Step 1

Rewrite the equation as $x^{3} \cdot 9^{3} = 9^{6}$ .

$x^{3} \cdot 9^{3} = 9^{6}$

Step 2

Divide each term in $x^{3} \cdot 9^{3} = 9^{6}$ by $9^{3}$ and simplify.

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Step 2.1

Divide each term in $x^{3} \cdot 9^{3} = 9^{6}$ by $9^{3}$ .

$\frac{x^{3} \cdot 9^{3}}{9^{3}} = \frac{9^{6}}{9^{3}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $9^{3}$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{x^{3} \cdot 9^{3}}{9^{3}} = \frac{9^{6}}{9^{3}}$

Step 2.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{9^{6}}{9^{3}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $9^{6}$ and $9^{3}$ .

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Step 2.3.1.1

Factor $9^{3}$ out of $9^{6}$ .

$x^{3} = \frac{9^{3} \cdot 9^{3}}{9^{3}}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Multiply by $1$ .

$x^{3} = \frac{9^{3} \cdot 9^{3}}{9^{3} \cdot 1}$

Step 2.3.1.2.2

Cancel the common factor.

$x^{3} = \frac{9^{3} \cdot 9^{3}}{9^{3} \cdot 1}$

Step 2.3.1.2.3

Rewrite the expression.

$x^{3} = \frac{9^{3}}{1}$

Step 2.3.1.2.4

Divide $9^{3}$ by $1$ .

$x^{3} = 9^{3}$

Step 2.3.2

Raise $9$ to the power of $3$ .

$x^{3} = 729$

Step 3

Subtract $729$ from both sides of the equation.

$x^{3} - 729 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

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

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

Step 4.2

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 $a = x$ and $b = 9$ .

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

Step 4.3

Simplify.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 5

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

$x - 9 = 0$

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

Step 6

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

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Step 6.1

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

$x - 9 = 0$

Step 6.2

Add $9$ to both sides of the equation.

$x = 9$

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

Use the quadratic formula to find the solutions.

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

Step 7.2.2

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

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

Step 7.2.3

Simplify.

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Step 7.2.3.1

Simplify the numerator.

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Step 7.2.3.1.1

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

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

Step 7.2.3.1.2

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

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Step 7.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.2.3.1.2.2

Multiply $- 4$ by $81$ .

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

Step 7.2.3.1.3

Subtract $324$ from $81$ .

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

Step 7.2.3.1.4

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

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

Step 7.2.3.1.5

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

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

Step 7.2.3.1.6

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

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

Step 7.2.3.1.7

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

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Step 7.2.3.1.7.1

Factor $81$ out of $243$ .

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

Step 7.2.3.1.7.2

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

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

Step 7.2.3.1.8

Pull terms out from under the radical.

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

Step 7.2.3.1.9

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

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

Step 7.2.3.2

Multiply $2$ by $1$ .

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

Step 7.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.2.4.1

Simplify the numerator.

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Step 7.2.4.1.1

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

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

Step 7.2.4.1.2

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

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Step 7.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.2.4.1.2.2

Multiply $- 4$ by $81$ .

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

Step 7.2.4.1.3

Subtract $324$ from $81$ .

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

Step 7.2.4.1.4

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

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

Step 7.2.4.1.5

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

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

Step 7.2.4.1.6

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

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

Step 7.2.4.1.7

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

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Step 7.2.4.1.7.1

Factor $81$ out of $243$ .

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

Step 7.2.4.1.7.2

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

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

Step 7.2.4.1.8

Pull terms out from under the radical.

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

Step 7.2.4.1.9

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

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

Step 7.2.4.2

Multiply $2$ by $1$ .

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

Step 7.2.4.3

Change the $\pm$ to $+$ .

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

Step 7.2.4.4

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 + 9 i \sqrt{3}}{2}$

Step 7.2.4.5

Factor $- 1$ out of $9 i \sqrt{3}$ .

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

Step 7.2.4.6

Factor $- 1$ out of $- 1 (9) - (- 9 i \sqrt{3})$ .

$x = \frac{- 1 (9 - 9 i \sqrt{3})}{2}$

Step 7.2.4.7

Move the negative in front of the fraction.

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

Step 7.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.2.5.1

Simplify the numerator.

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Step 7.2.5.1.1

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

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

Step 7.2.5.1.2

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

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Step 7.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.2.5.1.2.2

Multiply $- 4$ by $81$ .

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

Step 7.2.5.1.3

Subtract $324$ from $81$ .

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

Step 7.2.5.1.4

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

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

Step 7.2.5.1.5

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

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

Step 7.2.5.1.6

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

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

Step 7.2.5.1.7

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

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Step 7.2.5.1.7.1

Factor $81$ out of $243$ .

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

Step 7.2.5.1.7.2

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

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

Step 7.2.5.1.8

Pull terms out from under the radical.

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

Step 7.2.5.1.9

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

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

Step 7.2.5.2

Multiply $2$ by $1$ .

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

Step 7.2.5.3

Change the $\pm$ to $-$ .

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

Step 7.2.5.4

Rewrite $- 9$ as $- 1 (9)$ .

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

Step 7.2.5.5

Factor $- 1$ out of $- 9 i \sqrt{3}$ .

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

Step 7.2.5.6

Factor $- 1$ out of $- 1 (9) - (9 i \sqrt{3})$ .

$x = \frac{- 1 (9 + 9 i \sqrt{3})}{2}$

Step 7.2.5.7

Move the negative in front of the fraction.

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

Step 7.2.6

The final answer is the combination of both solutions.

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

Step 8

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

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