Algebra Examples

$x^{\frac{3}{2}} = 27$

Step 1

Subtract $27$ from both sides of the equation.

$x^{\frac{3}{2}} - 27 = 0$

Step 2

Rewrite $x^{\frac{3}{2}}$ as ${(x^{\frac{1}{2}})}^{3}$ .

${(x^{\frac{1}{2}})}^{3} - 27 = 0$

Step 3

Rewrite $27$ as $3^{3}$ .

${(x^{\frac{1}{2}})}^{3} - 3^{3} = 0$

Step 4

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 $a = x^{\frac{1}{2}}$ and $b = 3$ .

$(x^{\frac{1}{2}} - 3) ({(x^{\frac{1}{2}})}^{2} + x^{\frac{1}{2}} \cdot 3 + 3^{2}) = 0$

Step 5

Simplify.

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Step 5.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 5.1.1

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

$(x^{\frac{1}{2}} - 3) (x^{\frac{1}{2} \cdot 2} + x^{\frac{1}{2}} \cdot 3 + 3^{2}) = 0$

Step 5.1.2

Cancel the common factor of $2$ .

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Step 5.1.2.1

Cancel the common factor.

$(x^{\frac{1}{2}} - 3) (x^{\frac{1}{2} \cdot 2} + x^{\frac{1}{2}} \cdot 3 + 3^{2}) = 0$

Step 5.1.2.2

Rewrite the expression.

$(x^{\frac{1}{2}} - 3) (x + x^{\frac{1}{2}} \cdot 3 + 3^{2}) = 0$

Step 5.2

Simplify.

$(x^{\frac{1}{2}} - 3) (x + x^{\frac{1}{2}} \cdot 3 + 3^{2}) = 0$

Step 5.3

Move $3$ to the left of $x^{\frac{1}{2}}$ .

$(x^{\frac{1}{2}} - 3) (x + 3 x^{\frac{1}{2}} + 3^{2}) = 0$

Step 5.4

Raise $3$ to the power of $2$ .

$(x^{\frac{1}{2}} - 3) (x + 3 x^{\frac{1}{2}} + 9) = 0$

Step 6

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

$x^{\frac{1}{2}} - 3 = 0$

$x + 3 x^{\frac{1}{2}} + 9 = 0$

Step 7

Set $x^{\frac{1}{2}} - 3$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x^{\frac{1}{2}} - 3$ equal to $0$ .

$x^{\frac{1}{2}} - 3 = 0$

Step 7.2

Solve $x^{\frac{1}{2}} - 3 = 0$ for $x$ .

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Step 7.2.1

Add $3$ to both sides of the equation.

$x^{\frac{1}{2}} = 3$

Step 7.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = 3^{2}$

Step 7.2.3

Simplify the exponent.

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Step 7.2.3.1

Simplify the left side.

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Step 7.2.3.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 7.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 7.2.3.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 7.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 7.2.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 7.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 3^{2}$

Step 7.2.3.1.1.2

Simplify.

$x = 3^{2}$

Step 7.2.3.2

Simplify the right side.

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Step 7.2.3.2.1

Raise $3$ to the power of $2$ .

$x = 9$

Step 8

Set $x + 3 x^{\frac{1}{2}} + 9$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 3 x^{\frac{1}{2}} + 9$ equal to $0$ .

$x + 3 x^{\frac{1}{2}} + 9 = 0$

Step 8.2

Solve $x + 3 x^{\frac{1}{2}} + 9 = 0$ for $x$ .

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Step 8.2.1

Find a common factor $x^{\frac{1}{2}}$ that is present in each term.

${(x^{\frac{1}{2}})}^{2} + 3 x^{\frac{1}{2}} + 9$

Step 8.2.2

Substitute $u$ for $x^{\frac{1}{2}}$ .

${(u)}^{2} + 3 (u) + 9 = 0$

Step 8.2.3

Solve for $u$ .

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Step 8.2.3.1

Multiply $3$ by $u$ .

$u^{2} + 3 u + 9 = 0$

Step 8.2.3.2

Use the quadratic formula to find the solutions.

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

Step 8.2.3.3

Substitute the values $a = 1$ , $b = 3$ , and $c = 9$ into the quadratic formula and solve for $u$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 8.2.3.4

Simplify.

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Step 8.2.3.4.1

Simplify the numerator.

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Step 8.2.3.4.1.1

Raise $3$ to the power of $2$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 8.2.3.4.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 8.2.3.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 8.2.3.4.1.2.2

Multiply $- 4$ by $9$ .

$u = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 8.2.3.4.1.3

Subtract $36$ from $9$ .

$u = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 8.2.3.4.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$u = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 8.2.3.4.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$u = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 8.2.3.4.1.6

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

$u = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 8.2.3.4.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 8.2.3.4.1.7.1

Factor $9$ out of $27$ .

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

Step 8.2.3.4.1.7.2

Rewrite $9$ as $3^{2}$ .

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

Step 8.2.3.4.1.8

Pull terms out from under the radical.

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

Step 8.2.3.4.1.9

Move $3$ to the left of $i$ .

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

Step 8.2.3.4.2

Multiply $2$ by $1$ .

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

Step 8.2.3.5

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

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Step 8.2.3.5.1

Simplify the numerator.

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Step 8.2.3.5.1.1

Raise $3$ to the power of $2$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 8.2.3.5.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 8.2.3.5.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 8.2.3.5.1.2.2

Multiply $- 4$ by $9$ .

$u = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 8.2.3.5.1.3

Subtract $36$ from $9$ .

$u = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 8.2.3.5.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$u = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 8.2.3.5.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$u = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 8.2.3.5.1.6

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

$u = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 8.2.3.5.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 8.2.3.5.1.7.1

Factor $9$ out of $27$ .

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

Step 8.2.3.5.1.7.2

Rewrite $9$ as $3^{2}$ .

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

Step 8.2.3.5.1.8

Pull terms out from under the radical.

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

Step 8.2.3.5.1.9

Move $3$ to the left of $i$ .

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

Step 8.2.3.5.2

Multiply $2$ by $1$ .

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

Step 8.2.3.5.3

Change the $\pm$ to $+$ .

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

Step 8.2.3.5.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 8.2.3.5.5

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

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

Step 8.2.3.5.6

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

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

Step 8.2.3.5.7

Move the negative in front of the fraction.

$u = - \frac{3 - 3 i \sqrt{3}}{2}$

Step 8.2.3.6

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

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Step 8.2.3.6.1

Simplify the numerator.

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Step 8.2.3.6.1.1

Raise $3$ to the power of $2$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 8.2.3.6.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 8.2.3.6.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 8.2.3.6.1.2.2

Multiply $- 4$ by $9$ .

$u = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 8.2.3.6.1.3

Subtract $36$ from $9$ .

$u = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 8.2.3.6.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$u = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 8.2.3.6.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$u = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 8.2.3.6.1.6

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

$u = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 8.2.3.6.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 8.2.3.6.1.7.1

Factor $9$ out of $27$ .

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

Step 8.2.3.6.1.7.2

Rewrite $9$ as $3^{2}$ .

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

Step 8.2.3.6.1.8

Pull terms out from under the radical.

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

Step 8.2.3.6.1.9

Move $3$ to the left of $i$ .

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

Step 8.2.3.6.2

Multiply $2$ by $1$ .

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

Step 8.2.3.6.3

Change the $\pm$ to $-$ .

$u = \frac{- 3 - 3 i \sqrt{3}}{2}$

Step 8.2.3.6.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 8.2.3.6.5

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

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

Step 8.2.3.6.6

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

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

Step 8.2.3.6.7

Move the negative in front of the fraction.

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

Step 8.2.3.7

The final answer is the combination of both solutions.

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

Step 8.2.4

Substitute $x$ for $u$ .

$x^{\frac{1}{2}} = - \frac{3 - 3 i \sqrt{3}}{2}, - \frac{3 + 3 i \sqrt{3}}{2}$

Step 8.2.5

Solve for $x^{\frac{1}{2}} = - \frac{3 - 3 i \sqrt{3}}{2}$ for $x$ .

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Step 8.2.5.1

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 8.2.5.2

Simplify the exponent.

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Step 8.2.5.2.1

Simplify the left side.

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Step 8.2.5.2.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 8.2.5.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 8.2.5.2.1.1.1.1

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

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

Step 8.2.5.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 8.2.5.2.1.1.1.2.1

Cancel the common factor.

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

Step 8.2.5.2.1.1.1.2.2

Rewrite the expression.

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

Step 8.2.5.2.1.1.2

Simplify.

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

Step 8.2.5.2.2

Simplify the right side.

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Step 8.2.5.2.2.1

Simplify ${(- \frac{3 - 3 i \sqrt{3}}{2})}^{2}$ .

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Step 8.2.5.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.2.5.2.2.1.1.1

Apply the product rule to $- \frac{3 - 3 i \sqrt{3}}{2}$ .

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

Step 8.2.5.2.2.1.1.2

Apply the product rule to $\frac{3 - 3 i \sqrt{3}}{2}$ .

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

Step 8.2.5.2.2.1.2

Simplify the expression.

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Step 8.2.5.2.2.1.2.1

Raise $- 1$ to the power of $2$ .

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

Step 8.2.5.2.2.1.2.2

Multiply $\frac{{(3 - 3 i \sqrt{3})}^{2}}{2^{2}}$ by $1$ .

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

Step 8.2.5.2.2.1.2.3

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

$x = \frac{{(3 - 3 i \sqrt{3})}^{2}}{4}$

Step 8.2.5.2.2.1.2.4

Rewrite ${(3 - 3 i \sqrt{3})}^{2}$ as $(3 - 3 i \sqrt{3}) (3 - 3 i \sqrt{3})$ .

$x = \frac{(3 - 3 i \sqrt{3}) (3 - 3 i \sqrt{3})}{4}$

Step 8.2.5.2.2.1.3

Expand $(3 - 3 i \sqrt{3}) (3 - 3 i \sqrt{3})$ using the FOIL Method.

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Step 8.2.5.2.2.1.3.1

Apply the distributive property.

$x = \frac{3 (3 - 3 i \sqrt{3}) - 3 i \sqrt{3} (3 - 3 i \sqrt{3})}{4}$

Step 8.2.5.2.2.1.3.2

Apply the distributive property.

$x = \frac{3 \cdot 3 + 3 (- 3 i \sqrt{3}) - 3 i \sqrt{3} (3 - 3 i \sqrt{3})}{4}$

Step 8.2.5.2.2.1.3.3

Apply the distributive property.

$x = \frac{3 \cdot 3 + 3 (- 3 i \sqrt{3}) - 3 i \sqrt{3} \cdot 3 - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4}$

Step 8.2.5.2.2.1.4

Simplify and combine like terms.

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Step 8.2.5.2.2.1.4.1

Simplify each term.

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Step 8.2.5.2.2.1.4.1.1

Multiply $3$ by $3$ .

$x = \frac{9 + 3 (- 3 i \sqrt{3}) - 3 i \sqrt{3} \cdot 3 - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4}$

Step 8.2.5.2.2.1.4.1.2

Multiply $- 3$ by $3$ .

$x = \frac{9 - 9 (i \sqrt{3}) - 3 i \sqrt{3} \cdot 3 - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4}$

Step 8.2.5.2.2.1.4.1.3

Multiply $3$ by $- 3$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 3 i \sqrt{3} (- 3 i \sqrt{3})}{4}$

Step 8.2.5.2.2.1.4.1.4

Multiply $- 3 i \sqrt{3} (- 3 i \sqrt{3})$ .

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Step 8.2.5.2.2.1.4.1.4.1

Multiply $- 3$ by $- 3$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i \sqrt{3} (i \sqrt{3})}{4}$

Step 8.2.5.2.2.1.4.1.4.2

Raise $i$ to the power of $1$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 (i^{1} i) \sqrt{3} \sqrt{3}}{4}$

Step 8.2.5.2.2.1.4.1.4.3

Raise $i$ to the power of $1$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 (i^{1} i^{1}) \sqrt{3} \sqrt{3}}{4}$

Step 8.2.5.2.2.1.4.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{1 + 1} \sqrt{3} \sqrt{3}}{4}$

Step 8.2.5.2.2.1.4.1.4.5

Add $1$ and $1$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} \sqrt{3} \sqrt{3}}{4}$

Step 8.2.5.2.2.1.4.1.4.6

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{4}$

Step 8.2.5.2.2.1.4.1.4.7

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4}$

Step 8.2.5.2.2.1.4.1.4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{1 + 1}}{4}$

Step 8.2.5.2.2.1.4.1.4.9

Add $1$ and $1$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{2}}{4}$

Step 8.2.5.2.2.1.4.1.5

Rewrite $i^{2}$ as $- 1$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} + 9 \cdot - 1 {\sqrt{3}}^{2}}{4}$

Step 8.2.5.2.2.1.4.1.6

Multiply $9$ by $- 1$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 {\sqrt{3}}^{2}}{4}$

Step 8.2.5.2.2.1.4.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.2.5.2.2.1.4.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 {(3^{\frac{1}{2}})}^{2}}{4}$

Step 8.2.5.2.2.1.4.1.7.2

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

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 8.2.5.2.2.1.4.1.7.3

Combine $\frac{1}{2}$ and $2$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4}$

Step 8.2.5.2.2.1.4.1.7.4

Cancel the common factor of $2$ .

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Step 8.2.5.2.2.1.4.1.7.4.1

Cancel the common factor.

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4}$

Step 8.2.5.2.2.1.4.1.7.4.2

Rewrite the expression.

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3^{1}}{4}$

Step 8.2.5.2.2.1.4.1.7.5

Evaluate the exponent.

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 9 \cdot 3}{4}$

Step 8.2.5.2.2.1.4.1.8

Multiply $- 9$ by $3$ .

$x = \frac{9 - 9 i \sqrt{3} - 9 i \sqrt{3} - 27}{4}$

Step 8.2.5.2.2.1.4.2

Subtract $27$ from $9$ .

$x = \frac{- 9 i \sqrt{3} - 9 i \sqrt{3} - 18}{4}$

Step 8.2.5.2.2.1.4.3

Subtract $9 i \sqrt{3}$ from $- 9 i \sqrt{3}$ .

$x = \frac{- 18 i \sqrt{3} - 18}{4}$

Step 8.2.5.2.2.1.5

Reorder $- 18 i \sqrt{3}$ and $- 18$ .

$x = \frac{- 18 - 18 i \sqrt{3}}{4}$

Step 8.2.5.2.2.1.6

Cancel the common factor of $- 18 - 18 i \sqrt{3}$ and $4$ .

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Step 8.2.5.2.2.1.6.1

Factor $2$ out of $- 18$ .

$x = \frac{2 (- 9) - 18 i \sqrt{3}}{4}$

Step 8.2.5.2.2.1.6.2

Factor $2$ out of $- 18 i \sqrt{3}$ .

$x = \frac{2 (- 9) + 2 (- 9 i \sqrt{3})}{4}$

Step 8.2.5.2.2.1.6.3

Factor $2$ out of $2 (- 9) + 2 (- 9 i \sqrt{3})$ .

$x = \frac{2 (- 9 - 9 i \sqrt{3})}{4}$

Step 8.2.5.2.2.1.6.4

Cancel the common factors.

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Step 8.2.5.2.2.1.6.4.1

Factor $2$ out of $4$ .

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

Step 8.2.5.2.2.1.6.4.2

Cancel the common factor.

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

Step 8.2.5.2.2.1.6.4.3

Rewrite the expression.

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

Step 8.2.5.2.2.1.7

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

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

Step 8.2.5.2.2.1.8

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

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

Step 8.2.5.2.2.1.9

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

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

Step 8.2.5.2.2.1.10

Move the negative in front of the fraction.

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

Step 8.2.6

Solve for $x^{\frac{1}{2}} = - \frac{3 + 3 i \sqrt{3}}{2}$ for $x$ .

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Step 8.2.6.1

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 8.2.6.2

Simplify the exponent.

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Step 8.2.6.2.1

Simplify the left side.

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Step 8.2.6.2.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 8.2.6.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 8.2.6.2.1.1.1.1

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

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

Step 8.2.6.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 8.2.6.2.1.1.1.2.1

Cancel the common factor.

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

Step 8.2.6.2.1.1.1.2.2

Rewrite the expression.

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

Step 8.2.6.2.1.1.2

Simplify.

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

Step 8.2.6.2.2

Simplify the right side.

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Step 8.2.6.2.2.1

Simplify ${(- \frac{3 + 3 i \sqrt{3}}{2})}^{2}$ .

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Step 8.2.6.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.2.6.2.2.1.1.1

Apply the product rule to $- \frac{3 + 3 i \sqrt{3}}{2}$ .

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

Step 8.2.6.2.2.1.1.2

Apply the product rule to $\frac{3 + 3 i \sqrt{3}}{2}$ .

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

Step 8.2.6.2.2.1.2

Simplify the expression.

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Step 8.2.6.2.2.1.2.1

Raise $- 1$ to the power of $2$ .

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

Step 8.2.6.2.2.1.2.2

Multiply $\frac{{(3 + 3 i \sqrt{3})}^{2}}{2^{2}}$ by $1$ .

$x = \frac{{(3 + 3 i \sqrt{3})}^{2}}{2^{2}}$

Step 8.2.6.2.2.1.2.3

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

$x = \frac{{(3 + 3 i \sqrt{3})}^{2}}{4}$

Step 8.2.6.2.2.1.2.4

Rewrite ${(3 + 3 i \sqrt{3})}^{2}$ as $(3 + 3 i \sqrt{3}) (3 + 3 i \sqrt{3})$ .

$x = \frac{(3 + 3 i \sqrt{3}) (3 + 3 i \sqrt{3})}{4}$

Step 8.2.6.2.2.1.3

Expand $(3 + 3 i \sqrt{3}) (3 + 3 i \sqrt{3})$ using the FOIL Method.

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Step 8.2.6.2.2.1.3.1

Apply the distributive property.

$x = \frac{3 (3 + 3 i \sqrt{3}) + 3 i \sqrt{3} (3 + 3 i \sqrt{3})}{4}$

Step 8.2.6.2.2.1.3.2

Apply the distributive property.

$x = \frac{3 \cdot 3 + 3 (3 i \sqrt{3}) + 3 i \sqrt{3} (3 + 3 i \sqrt{3})}{4}$

Step 8.2.6.2.2.1.3.3

Apply the distributive property.

$x = \frac{3 \cdot 3 + 3 (3 i \sqrt{3}) + 3 i \sqrt{3} \cdot 3 + 3 i \sqrt{3} (3 i \sqrt{3})}{4}$

Step 8.2.6.2.2.1.4

Simplify and combine like terms.

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Step 8.2.6.2.2.1.4.1

Simplify each term.

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Step 8.2.6.2.2.1.4.1.1

Multiply $3$ by $3$ .

$x = \frac{9 + 3 (3 i \sqrt{3}) + 3 i \sqrt{3} \cdot 3 + 3 i \sqrt{3} (3 i \sqrt{3})}{4}$

Step 8.2.6.2.2.1.4.1.2

Multiply $3$ by $3$ .

$x = \frac{9 + 9 (i \sqrt{3}) + 3 i \sqrt{3} \cdot 3 + 3 i \sqrt{3} (3 i \sqrt{3})}{4}$

Step 8.2.6.2.2.1.4.1.3

Multiply $3$ by $3$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 3 i \sqrt{3} (3 i \sqrt{3})}{4}$

Step 8.2.6.2.2.1.4.1.4

Multiply $3 i \sqrt{3} (3 i \sqrt{3})$ .

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Step 8.2.6.2.2.1.4.1.4.1

Multiply $3$ by $3$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i \sqrt{3} (i \sqrt{3})}{4}$

Step 8.2.6.2.2.1.4.1.4.2

Raise $i$ to the power of $1$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 (i^{1} i) \sqrt{3} \sqrt{3}}{4}$

Step 8.2.6.2.2.1.4.1.4.3

Raise $i$ to the power of $1$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 (i^{1} i^{1}) \sqrt{3} \sqrt{3}}{4}$

Step 8.2.6.2.2.1.4.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{1 + 1} \sqrt{3} \sqrt{3}}{4}$

Step 8.2.6.2.2.1.4.1.4.5

Add $1$ and $1$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{2} \sqrt{3} \sqrt{3}}{4}$

Step 8.2.6.2.2.1.4.1.4.6

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{4}$

Step 8.2.6.2.2.1.4.1.4.7

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4}$

Step 8.2.6.2.2.1.4.1.4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{1 + 1}}{4}$

Step 8.2.6.2.2.1.4.1.4.9

Add $1$ and $1$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{2}}{4}$

Step 8.2.6.2.2.1.4.1.5

Rewrite $i^{2}$ as $- 1$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} + 9 \cdot - 1 {\sqrt{3}}^{2}}{4}$

Step 8.2.6.2.2.1.4.1.6

Multiply $9$ by $- 1$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 {\sqrt{3}}^{2}}{4}$

Step 8.2.6.2.2.1.4.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.2.6.2.2.1.4.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 {(3^{\frac{1}{2}})}^{2}}{4}$

Step 8.2.6.2.2.1.4.1.7.2

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

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 8.2.6.2.2.1.4.1.7.3

Combine $\frac{1}{2}$ and $2$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4}$

Step 8.2.6.2.2.1.4.1.7.4

Cancel the common factor of $2$ .

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Step 8.2.6.2.2.1.4.1.7.4.1

Cancel the common factor.

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4}$

Step 8.2.6.2.2.1.4.1.7.4.2

Rewrite the expression.

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3^{1}}{4}$

Step 8.2.6.2.2.1.4.1.7.5

Evaluate the exponent.

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 9 \cdot 3}{4}$

Step 8.2.6.2.2.1.4.1.8

Multiply $- 9$ by $3$ .

$x = \frac{9 + 9 i \sqrt{3} + 9 i \sqrt{3} - 27}{4}$

Step 8.2.6.2.2.1.4.2

Subtract $27$ from $9$ .

$x = \frac{9 i \sqrt{3} + 9 i \sqrt{3} - 18}{4}$

Step 8.2.6.2.2.1.4.3

Add $9 i \sqrt{3}$ and $9 i \sqrt{3}$ .

$x = \frac{18 i \sqrt{3} - 18}{4}$

Step 8.2.6.2.2.1.5

Reorder $18 i \sqrt{3}$ and $- 18$ .

$x = \frac{- 18 + 18 i \sqrt{3}}{4}$

Step 8.2.6.2.2.1.6

Cancel the common factor of $- 18 + 18 i \sqrt{3}$ and $4$ .

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Step 8.2.6.2.2.1.6.1

Factor $2$ out of $- 18$ .

$x = \frac{2 (- 9) + 18 i \sqrt{3}}{4}$

Step 8.2.6.2.2.1.6.2

Factor $2$ out of $18 i \sqrt{3}$ .

$x = \frac{2 (- 9) + 2 (9 i \sqrt{3})}{4}$

Step 8.2.6.2.2.1.6.3

Factor $2$ out of $2 (- 9) + 2 (9 i \sqrt{3})$ .

$x = \frac{2 (- 9 + 9 i \sqrt{3})}{4}$

Step 8.2.6.2.2.1.6.4

Cancel the common factors.

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Step 8.2.6.2.2.1.6.4.1

Factor $2$ out of $4$ .

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

Step 8.2.6.2.2.1.6.4.2

Cancel the common factor.

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

Step 8.2.6.2.2.1.6.4.3

Rewrite the expression.

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

Step 8.2.6.2.2.1.7

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

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

Step 8.2.6.2.2.1.8

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

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

Step 8.2.6.2.2.1.9

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

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

Step 8.2.6.2.2.1.10

Move the negative in front of the fraction.

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

Step 8.2.7

List all of the solutions.

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

Step 9

The final solution is all the values that make $(x^{\frac{1}{2}} - 3) (x + 3 x^{\frac{1}{2}} + 9) = 0$ true.

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