Algebra Examples

$12 x^{\frac{7}{2}} - 108 x^{3} = 0$

Step 1

Find a common factor $x^{3}$ that is present in each term.

$12 {(x^{3})}^{\frac{7}{6}} - 108 x^{3}$

Step 2

Substitute $u$ for $x^{3}$ .

$12 {(u)}^{\frac{7}{6}} - 108 u = 0$

Step 3

Solve for $u$ .

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Step 3.1

Factor $12 u$ out of $12 u^{\frac{7}{6}} - 108 u$ .

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Step 3.1.1

Factor $12 u$ out of $12 u^{\frac{7}{6}}$ .

$12 u (u^{\frac{1}{6}}) - 108 u = 0$

Step 3.1.2

Factor $12 u$ out of $- 108 u$ .

$12 u (u^{\frac{1}{6}}) + 12 u (- 9) = 0$

Step 3.1.3

Factor $12 u$ out of $12 u (u^{\frac{1}{6}}) + 12 u (- 9)$ .

$12 u (u^{\frac{1}{6}} - 9) = 0$

Step 3.2

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

$u = 0$

$u^{\frac{1}{6}} - 9 = 0$

Step 3.3

Set $u$ equal to $0$ .

$u = 0$

Step 3.4

Set $u^{\frac{1}{6}} - 9$ equal to $0$ and solve for $u$ .

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Step 3.4.1

Set $u^{\frac{1}{6}} - 9$ equal to $0$ .

$u^{\frac{1}{6}} - 9 = 0$

Step 3.4.2

Solve $u^{\frac{1}{6}} - 9 = 0$ for $u$ .

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Step 3.4.2.1

Add $9$ to both sides of the equation.

$u^{\frac{1}{6}} = 9$

Step 3.4.2.2

Raise each side of the equation to the power of $6$ to eliminate the fractional exponent on the left side.

${(u^{\frac{1}{6}})}^{6} = 9^{6}$

Step 3.4.2.3

Simplify the exponent.

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Step 3.4.2.3.1

Simplify the left side.

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Step 3.4.2.3.1.1

Simplify ${(u^{\frac{1}{6}})}^{6}$ .

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Step 3.4.2.3.1.1.1

Multiply the exponents in ${(u^{\frac{1}{6}})}^{6}$ .

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Step 3.4.2.3.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$u^{\frac{1}{6} \cdot 6} = 9^{6}$

Step 3.4.2.3.1.1.1.2

Cancel the common factor of $6$ .

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Step 3.4.2.3.1.1.1.2.1

Cancel the common factor.

$u^{\frac{1}{6} \cdot 6} = 9^{6}$

Step 3.4.2.3.1.1.1.2.2

Rewrite the expression.

$u^{1} = 9^{6}$

Step 3.4.2.3.1.1.2

Simplify.

$u = 9^{6}$

Step 3.4.2.3.2

Simplify the right side.

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Step 3.4.2.3.2.1

Raise $9$ to the power of $6$ .

$u = 531441$

Step 3.5

The final solution is all the values that make $12 u (u^{\frac{1}{6}} - 9) = 0$ true.

$u = 0, 531441$

Step 4

Substitute $x$ for $u$ .

$x^{3} = 0, 531441$

Step 5

Solve for $x^{3} = 0$ for $x$ .

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Step 5.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 5.2

Simplify $\sqrt[3]{0}$ .

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Step 5.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 5.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 6

Solve for $x^{3} = 531441$ for $x$ .

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Step 6.1

Subtract $531441$ from both sides of the equation.

$x^{3} - 531441 = 0$

Step 6.2

Factor the left side of the equation.

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Step 6.2.1

Rewrite $531441$ as $81^{3}$ .

$x^{3} - 81^{3} = 0$

Step 6.2.2

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 $b = 81$ .

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

Step 6.2.3

Simplify.

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Step 6.2.3.1

Move $81$ to the left of $x$ .

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

Step 6.2.3.2

Raise $81$ to the power of $2$ .

$(x - 81) (x^{2} + 81 x + 6561) = 0$

Step 6.3

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

$x - 81 = 0$

$x^{2} + 81 x + 6561 = 0$

Step 6.4

Set $x - 81$ equal to $0$ and solve for $x$ .

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Step 6.4.1

Set $x - 81$ equal to $0$ .

$x - 81 = 0$

Step 6.4.2

Add $81$ to both sides of the equation.

$x = 81$

Step 6.5

Set $x^{2} + 81 x + 6561$ equal to $0$ and solve for $x$ .

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Step 6.5.1

Set $x^{2} + 81 x + 6561$ equal to $0$ .

$x^{2} + 81 x + 6561 = 0$

Step 6.5.2

Solve $x^{2} + 81 x + 6561 = 0$ for $x$ .

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Step 6.5.2.1

Use the quadratic formula to find the solutions.

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

Step 6.5.2.2

Substitute the values $a = 1$ , $b = 81$ , and $c = 6561$ into the quadratic formula and solve for $x$ .

$\frac{- 81 \pm \sqrt{81^{2} - 4 \cdot (1 \cdot 6561)}}{2 \cdot 1}$

Step 6.5.2.3

Simplify.

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Step 6.5.2.3.1

Simplify the numerator.

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Step 6.5.2.3.1.1

Raise $81$ to the power of $2$ .

$x = \frac{- 81 \pm \sqrt{6561 - 4 \cdot 1 \cdot 6561}}{2 \cdot 1}$

Step 6.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 6561$ .

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Step 6.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 81 \pm \sqrt{6561 - 4 \cdot 6561}}{2 \cdot 1}$

Step 6.5.2.3.1.2.2

Multiply $- 4$ by $6561$ .

$x = \frac{- 81 \pm \sqrt{6561 - 26244}}{2 \cdot 1}$

Step 6.5.2.3.1.3

Subtract $26244$ from $6561$ .

$x = \frac{- 81 \pm \sqrt{- 19683}}{2 \cdot 1}$

Step 6.5.2.3.1.4

Rewrite $- 19683$ as $- 1 (19683)$ .

$x = \frac{- 81 \pm \sqrt{- 1 \cdot 19683}}{2 \cdot 1}$

Step 6.5.2.3.1.5

Rewrite $\sqrt{- 1 (19683)}$ as $\sqrt{- 1} \cdot \sqrt{19683}$ .

$x = \frac{- 81 \pm \sqrt{- 1} \cdot \sqrt{19683}}{2 \cdot 1}$

Step 6.5.2.3.1.6

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

$x = \frac{- 81 \pm i \cdot \sqrt{19683}}{2 \cdot 1}$

Step 6.5.2.3.1.7

Rewrite $19683$ as $81^{2} \cdot 3$ .

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Step 6.5.2.3.1.7.1

Factor $6561$ out of $19683$ .

$x = \frac{- 81 \pm i \cdot \sqrt{6561 (3)}}{2 \cdot 1}$

Step 6.5.2.3.1.7.2

Rewrite $6561$ as $81^{2}$ .

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

Step 6.5.2.3.1.8

Pull terms out from under the radical.

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

Step 6.5.2.3.1.9

Move $81$ to the left of $i$ .

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

Step 6.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 81 \pm 81 i \sqrt{3}}{2}$

Step 6.5.2.4

The final answer is the combination of both solutions.

$x = - \frac{81 - 81 i \sqrt{3}}{2}, - \frac{81 + 81 i \sqrt{3}}{2}$

Step 6.6

The final solution is all the values that make $(x - 81) (x^{2} + 81 x + 6561) = 0$ true.

$x = 81, - \frac{81 - 81 i \sqrt{3}}{2}, - \frac{81 + 81 i \sqrt{3}}{2}$

Step 7

List all of the solutions.

$x = 0, 81, - \frac{81 - 81 i \sqrt{3}}{2}, - \frac{81 + 81 i \sqrt{3}}{2}$