Algebra Examples

$x^{- \frac{3}{2}} = \frac{1}{729}$

Step 1

Subtract $\frac{1}{729}$ from both sides of the equation.

$x^{- \frac{3}{2}} - \frac{1}{729} = 0$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{\frac{3}{2}}} - \frac{1}{729} = 0$

Step 3

Rewrite $1$ as $1^{3}$ .

$\frac{1^{3}}{x^{\frac{3}{2}}} - \frac{1}{729} = 0$

Step 4

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

$\frac{1^{3}}{{(x^{\frac{1}{2}})}^{3}} - \frac{1}{729} = 0$

Step 5

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

${(\frac{1}{x^{\frac{1}{2}}})}^{3} - \frac{1}{729} = 0$

Step 6

Rewrite $\frac{1}{729}$ as ${(\frac{1}{9})}^{3}$ .

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

Step 7

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 $a = \frac{1}{x^{\frac{1}{2}}}$ and $b = \frac{1}{9}$ .

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

Step 8

Simplify.

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Step 8.1

Apply the product rule to $\frac{1}{x^{\frac{1}{2}}}$ .

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

Step 8.2

One to any power is one.

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

Step 8.3

Simplify the denominator.

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Step 8.3.1

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

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Step 8.3.1.1

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

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

Step 8.3.1.2

Cancel the common factor of $2$ .

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Step 8.3.1.2.1

Cancel the common factor.

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

Step 8.3.1.2.2

Rewrite the expression.

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

Step 8.3.2

Simplify.

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

Step 8.4

Combine.

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

Step 8.5

Multiply $1$ by $1$ .

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

Step 8.6

Move $9$ to the left of $x^{\frac{1}{2}}$ .

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

Step 8.7

Multiply $9$ by $x^{\frac{1}{2}}$ .

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

Step 8.8

Apply the product rule to $\frac{1}{9}$ .

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

Step 8.9

One to any power is one.

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

Step 8.10

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

$(\frac{1}{x^{\frac{1}{2}}} - \frac{1}{9}) (\frac{1}{x} + \frac{1}{9 x^{\frac{1}{2}}} + \frac{1}{81}) = 0$

Step 9

To write $\frac{1}{x^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$(\frac{1}{x^{\frac{1}{2}}} \cdot \frac{9}{9} - \frac{1}{9}) (\frac{1}{x} + \frac{1}{9 x^{\frac{1}{2}}} + \frac{1}{81}) = 0$

Step 10

To write $- \frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

$(\frac{1}{x^{\frac{1}{2}}} \cdot \frac{9}{9} - \frac{1}{9} \cdot \frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}) (\frac{1}{x} + \frac{1}{9 x^{\frac{1}{2}}} + \frac{1}{81}) = 0$

Step 11

Write each expression with a common denominator of $9 x^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.1

Multiply $\frac{1}{x^{\frac{1}{2}}}$ by $\frac{9}{9}$ .

$(\frac{9}{x^{\frac{1}{2}} \cdot 9} - \frac{1}{9} \cdot \frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}) (\frac{1}{x} + \frac{1}{9 x^{\frac{1}{2}}} + \frac{1}{81}) = 0$

Step 11.2

Multiply $\frac{1}{9}$ by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

$(\frac{9}{x^{\frac{1}{2}} \cdot 9} - \frac{x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}}) (\frac{1}{x} + \frac{1}{9 x^{\frac{1}{2}}} + \frac{1}{81}) = 0$

Step 11.3

Reorder the factors of $x^{\frac{1}{2}} \cdot 9$ .

$(\frac{9}{9 x^{\frac{1}{2}}} - \frac{x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}}) (\frac{1}{x} + \frac{1}{9 x^{\frac{1}{2}}} + \frac{1}{81}) = 0$

Step 12

Combine the numerators over the common denominator.

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{1}{x} + \frac{1}{9 x^{\frac{1}{2}}} + \frac{1}{81}) = 0$

Step 13

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{9 x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{1}{x} \cdot \frac{9 x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} + \frac{1}{9 x^{\frac{1}{2}}} + \frac{1}{81}) = 0$

Step 14

To write $\frac{1}{9 x^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{1}{x} \cdot \frac{9 x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} + \frac{1}{9 x^{\frac{1}{2}}} \cdot \frac{x}{x} + \frac{1}{81}) = 0$

Step 15

Write each expression with a common denominator of $x \cdot 9 x^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.1

Multiply $\frac{1}{x}$ by $\frac{9 x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{x (9 x^{\frac{1}{2}})} + \frac{1}{9 x^{\frac{1}{2}}} \cdot \frac{x}{x} + \frac{1}{81}) = 0$

Step 15.2

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

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{9 (x \cdot x^{\frac{1}{2}})} + \frac{1}{9 x^{\frac{1}{2}}} \cdot \frac{x}{x} + \frac{1}{81}) = 0$

Step 15.3

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

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{9 x^{1 + \frac{1}{2}}} + \frac{1}{9 x^{\frac{1}{2}}} \cdot \frac{x}{x} + \frac{1}{81}) = 0$

Step 15.4

Write $1$ as a fraction with a common denominator.

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{9 x^{\frac{2}{2} + \frac{1}{2}}} + \frac{1}{9 x^{\frac{1}{2}}} \cdot \frac{x}{x} + \frac{1}{81}) = 0$

Step 15.5

Combine the numerators over the common denominator.

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{9 x^{\frac{2 + 1}{2}}} + \frac{1}{9 x^{\frac{1}{2}}} \cdot \frac{x}{x} + \frac{1}{81}) = 0$

Step 15.6

Add $2$ and $1$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{9 x^{\frac{3}{2}}} + \frac{1}{9 x^{\frac{1}{2}}} \cdot \frac{x}{x} + \frac{1}{81}) = 0$

Step 15.7

Multiply $\frac{1}{9 x^{\frac{1}{2}}}$ by $\frac{x}{x}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{9 x^{\frac{3}{2}}} + \frac{x}{9 x^{\frac{1}{2}} x} + \frac{1}{81}) = 0$

Step 15.8

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

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{9 x^{\frac{3}{2}}} + \frac{x}{9 (x \cdot x^{\frac{1}{2}})} + \frac{1}{81}) = 0$

Step 15.9

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

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{9 x^{\frac{3}{2}}} + \frac{x}{9 x^{1 + \frac{1}{2}}} + \frac{1}{81}) = 0$

Step 15.10

Write $1$ as a fraction with a common denominator.

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{9 x^{\frac{3}{2}}} + \frac{x}{9 x^{\frac{2}{2} + \frac{1}{2}}} + \frac{1}{81}) = 0$

Step 15.11

Combine the numerators over the common denominator.

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{9 x^{\frac{3}{2}}} + \frac{x}{9 x^{\frac{2 + 1}{2}}} + \frac{1}{81}) = 0$

Step 15.12

Add $2$ and $1$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}}}{9 x^{\frac{3}{2}}} + \frac{x}{9 x^{\frac{3}{2}}} + \frac{1}{81}) = 0$

Step 16

Combine the numerators over the common denominator.

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 x^{\frac{1}{2}} + x}{9 x^{\frac{3}{2}}} + \frac{1}{81}) = 0$

Step 17

Simplify each term.

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Step 17.1

Factor $x^{\frac{1}{2}}$ out of $9 x^{\frac{1}{2}} + x$ .

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Step 17.1.1

Factor $x^{\frac{1}{2}}$ out of $9 x^{\frac{1}{2}}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{x^{\frac{1}{2}} \cdot 9 + x}{9 x^{\frac{3}{2}}} + \frac{1}{81}) = 0$

Step 17.1.2

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

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{x^{\frac{1}{2}} \cdot 9 + x}{9 x^{\frac{3}{2}}} + \frac{1}{81}) = 0$

Step 17.1.3

Factor $x^{\frac{1}{2}}$ out of $x^{1}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{x^{\frac{1}{2}} \cdot 9 + x^{\frac{1}{2}} x^{\frac{1}{2}}}{9 x^{\frac{3}{2}}} + \frac{1}{81}) = 0$

Step 17.1.4

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 9 + x^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{x^{\frac{1}{2}} \cdot (9 + x^{\frac{1}{2}})}{9 x^{\frac{3}{2}}} + \frac{1}{81}) = 0$

Step 17.1.5

Multiply $x^{\frac{1}{2}}$ by $9 + x^{\frac{1}{2}}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{x^{\frac{1}{2}} (9 + x^{\frac{1}{2}})}{9 x^{\frac{3}{2}}} + \frac{1}{81}) = 0$

Step 17.2

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 + x^{\frac{1}{2}}}{9 x^{\frac{3}{2}} x^{- \frac{1}{2}}} + \frac{1}{81}) = 0$

Step 17.3

Simplify the denominator.

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Step 17.3.1

Multiply $x^{\frac{3}{2}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 17.3.1.1

Move $x^{- \frac{1}{2}}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 + x^{\frac{1}{2}}}{9 (x^{- \frac{1}{2}} x^{\frac{3}{2}})} + \frac{1}{81}) = 0$

Step 17.3.1.2

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

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 + x^{\frac{1}{2}}}{9 x^{- \frac{1}{2} + \frac{3}{2}}} + \frac{1}{81}) = 0$

Step 17.3.1.3

Combine the numerators over the common denominator.

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 + x^{\frac{1}{2}}}{9 x^{\frac{- 1 + 3}{2}}} + \frac{1}{81}) = 0$

Step 17.3.1.4

Add $- 1$ and $3$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 + x^{\frac{1}{2}}}{9 x^{\frac{2}{2}}} + \frac{1}{81}) = 0$

Step 17.3.1.5

Divide $2$ by $2$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 + x^{\frac{1}{2}}}{9 x} + \frac{1}{81}) = 0$

Step 17.3.2

Simplify $9 x^{1}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 + x^{\frac{1}{2}}}{9 x} + \frac{1}{81}) = 0$

Step 18

To write $\frac{9 + x^{\frac{1}{2}}}{9 x}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 + x^{\frac{1}{2}}}{9 x} \cdot \frac{9}{9} + \frac{1}{81}) = 0$

Step 19

To write $\frac{1}{81}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{9 + x^{\frac{1}{2}}}{9 x} \cdot \frac{9}{9} + \frac{1}{81} \cdot \frac{x}{x}) = 0$

Step 20

Write each expression with a common denominator of $81 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 20.1

Multiply $\frac{9 + x^{\frac{1}{2}}}{9 x}$ by $\frac{9}{9}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{(9 + x^{\frac{1}{2}}) \cdot 9}{9 x \cdot 9} + \frac{1}{81} \cdot \frac{x}{x}) = 0$

Step 20.2

Multiply $9$ by $9$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{(9 + x^{\frac{1}{2}}) \cdot 9}{81 x} + \frac{1}{81} \cdot \frac{x}{x}) = 0$

Step 20.3

Multiply $\frac{1}{81}$ by $\frac{x}{x}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot (\frac{(9 + x^{\frac{1}{2}}) \cdot 9}{81 x} + \frac{x}{81 x}) = 0$

Step 21

Combine the numerators over the common denominator.

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot \frac{(9 + x^{\frac{1}{2}}) \cdot 9 + x}{81 x} = 0$

Step 22

Simplify the numerator.

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Step 22.1

Apply the distributive property.

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot \frac{9 \cdot 9 + x^{\frac{1}{2}} \cdot 9 + x}{81 x} = 0$

Step 22.2

Multiply $9$ by $9$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot \frac{81 + x^{\frac{1}{2}} \cdot 9 + x}{81 x} = 0$

Step 22.3

Move $9$ to the left of $x^{\frac{1}{2}}$ .

$\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot \frac{81 + 9 x^{\frac{1}{2}} + x}{81 x} = 0$

Step 23

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

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

$\frac{81 + 9 x^{\frac{1}{2}} + x}{81 x} = 0$

Step 24

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

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Step 24.1

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

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

Step 24.2

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

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Step 24.2.1

Set the numerator equal to zero.

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

Step 24.2.2

Solve the equation for $x$ .

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Step 24.2.2.1

Subtract $9$ from both sides of the equation.

$- x^{\frac{1}{2}} = - 9$

Step 24.2.2.2

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

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

Step 24.2.2.3

Simplify the exponent.

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Step 24.2.2.3.1

Simplify the left side.

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Step 24.2.2.3.1.1

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

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Step 24.2.2.3.1.1.1

Apply the product rule to $- x^{\frac{1}{2}}$ .

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

Step 24.2.2.3.1.1.2

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

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

Step 24.2.2.3.1.1.3

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

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

Step 24.2.2.3.1.1.4

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

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Step 24.2.2.3.1.1.4.1

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

$x^{\frac{1}{2} \cdot 2} = {(- 9)}^{2}$

Step 24.2.2.3.1.1.4.2

Cancel the common factor of $2$ .

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Step 24.2.2.3.1.1.4.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(- 9)}^{2}$

Step 24.2.2.3.1.1.4.2.2

Rewrite the expression.

$x^{1} = {(- 9)}^{2}$

Step 24.2.2.3.1.1.5

Simplify.

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

Step 24.2.2.3.2

Simplify the right side.

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Step 24.2.2.3.2.1

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

$x = 81$

Step 25

Set $\frac{81 + 9 x^{\frac{1}{2}} + x}{81 x}$ equal to $0$ and solve for $x$ .

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Step 25.1

Set $\frac{81 + 9 x^{\frac{1}{2}} + x}{81 x}$ equal to $0$ .

$\frac{81 + 9 x^{\frac{1}{2}} + x}{81 x} = 0$

Step 25.2

Solve $\frac{81 + 9 x^{\frac{1}{2}} + x}{81 x} = 0$ for $x$ .

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Step 25.2.1

Multiply both sides by $81 x$ .

$\frac{81 + 9 x^{\frac{1}{2}} + x}{81 x} (81 x) = 0 (81 x)$

Step 25.2.2

Simplify.

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Step 25.2.2.1

Simplify the left side.

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Step 25.2.2.1.1

Simplify $\frac{81 + 9 x^{\frac{1}{2}} + x}{81 x} (81 x)$ .

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Step 25.2.2.1.1.1

Rewrite using the commutative property of multiplication.

$81 \frac{81 + 9 x^{\frac{1}{2}} + x}{81 x} x = 0 (81 x)$

Step 25.2.2.1.1.2

Cancel the common factor of $81$ .

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Step 25.2.2.1.1.2.1

Factor $81$ out of $81 x$ .

$81 \frac{81 + 9 x^{\frac{1}{2}} + x}{81 (x)} x = 0 (81 x)$

Step 25.2.2.1.1.2.2

Cancel the common factor.

$81 \frac{81 + 9 x^{\frac{1}{2}} + x}{81 x} x = 0 (81 x)$

Step 25.2.2.1.1.2.3

Rewrite the expression.

$\frac{81 + 9 x^{\frac{1}{2}} + x}{x} x = 0 (81 x)$

Step 25.2.2.1.1.3

Cancel the common factor of $x$ .

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Step 25.2.2.1.1.3.1

Cancel the common factor.

$\frac{81 + 9 x^{\frac{1}{2}} + x}{x} x = 0 (81 x)$

Step 25.2.2.1.1.3.2

Rewrite the expression.

$81 + 9 x^{\frac{1}{2}} + x = 0 (81 x)$

Step 25.2.2.1.1.4

Simplify the expression.

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Step 25.2.2.1.1.4.1

Move $9 x^{\frac{1}{2}}$ .

$81 + x + 9 x^{\frac{1}{2}} = 0 (81 x)$

Step 25.2.2.1.1.4.2

Reorder $81$ and $x$ .

$x + 81 + 9 x^{\frac{1}{2}} = 0 (81 x)$

Step 25.2.2.2

Simplify the right side.

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Step 25.2.2.2.1

Multiply $0 (81 x)$ .

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Step 25.2.2.2.1.1

Multiply $81$ by $0$ .

$x + 81 + 9 x^{\frac{1}{2}} = 0 x$

Step 25.2.2.2.1.2

Multiply $0$ by $x$ .

$x + 81 + 9 x^{\frac{1}{2}} = 0$

Step 25.2.3

Solve for $x$ .

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Step 25.2.3.1

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

${(x^{\frac{1}{2}})}^{2} + 81 + 9 x^{\frac{1}{2}}$

Step 25.2.3.2

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

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

Step 25.2.3.3

Solve for $u$ .

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Step 25.2.3.3.1

Multiply $9$ by $u$ .

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

Step 25.2.3.3.2

Use the quadratic formula to find the solutions.

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

Step 25.2.3.3.3

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

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

Step 25.2.3.3.4

Simplify.

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Step 25.2.3.3.4.1

Simplify the numerator.

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Step 25.2.3.3.4.1.1

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

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

Step 25.2.3.3.4.1.2

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

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Step 25.2.3.3.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 25.2.3.3.4.1.2.2

Multiply $- 4$ by $81$ .

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

Step 25.2.3.3.4.1.3

Subtract $324$ from $81$ .

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

Step 25.2.3.3.4.1.4

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

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

Step 25.2.3.3.4.1.5

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

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

Step 25.2.3.3.4.1.6

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

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

Step 25.2.3.3.4.1.7

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

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Step 25.2.3.3.4.1.7.1

Factor $81$ out of $243$ .

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

Step 25.2.3.3.4.1.7.2

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

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

Step 25.2.3.3.4.1.8

Pull terms out from under the radical.

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

Step 25.2.3.3.4.1.9

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

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

Step 25.2.3.3.4.2

Multiply $2$ by $1$ .

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

Step 25.2.3.3.5

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

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Step 25.2.3.3.5.1

Simplify the numerator.

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Step 25.2.3.3.5.1.1

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

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

Step 25.2.3.3.5.1.2

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

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Step 25.2.3.3.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 25.2.3.3.5.1.2.2

Multiply $- 4$ by $81$ .

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

Step 25.2.3.3.5.1.3

Subtract $324$ from $81$ .

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

Step 25.2.3.3.5.1.4

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

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

Step 25.2.3.3.5.1.5

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

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

Step 25.2.3.3.5.1.6

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

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

Step 25.2.3.3.5.1.7

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

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Step 25.2.3.3.5.1.7.1

Factor $81$ out of $243$ .

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

Step 25.2.3.3.5.1.7.2

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

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

Step 25.2.3.3.5.1.8

Pull terms out from under the radical.

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

Step 25.2.3.3.5.1.9

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

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

Step 25.2.3.3.5.2

Multiply $2$ by $1$ .

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

Step 25.2.3.3.5.3

Change the $\pm$ to $+$ .

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

Step 25.2.3.3.5.4

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

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

Step 25.2.3.3.5.5

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

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

Step 25.2.3.3.5.6

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

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

Step 25.2.3.3.5.7

Move the negative in front of the fraction.

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

Step 25.2.3.3.6

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

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Step 25.2.3.3.6.1

Simplify the numerator.

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Step 25.2.3.3.6.1.1

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

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

Step 25.2.3.3.6.1.2

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

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Step 25.2.3.3.6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 25.2.3.3.6.1.2.2

Multiply $- 4$ by $81$ .

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

Step 25.2.3.3.6.1.3

Subtract $324$ from $81$ .

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

Step 25.2.3.3.6.1.4

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

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

Step 25.2.3.3.6.1.5

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

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

Step 25.2.3.3.6.1.6

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

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

Step 25.2.3.3.6.1.7

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

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Step 25.2.3.3.6.1.7.1

Factor $81$ out of $243$ .

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

Step 25.2.3.3.6.1.7.2

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

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

Step 25.2.3.3.6.1.8

Pull terms out from under the radical.

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

Step 25.2.3.3.6.1.9

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

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

Step 25.2.3.3.6.2

Multiply $2$ by $1$ .

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

Step 25.2.3.3.6.3

Change the $\pm$ to $-$ .

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

Step 25.2.3.3.6.4

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

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

Step 25.2.3.3.6.5

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

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

Step 25.2.3.3.6.6

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

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

Step 25.2.3.3.6.7

Move the negative in front of the fraction.

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

Step 25.2.3.3.7

The final answer is the combination of both solutions.

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

Step 25.2.3.4

Substitute $x$ for $u$ .

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

Step 25.2.3.5

Solve for $x^{\frac{1}{2}} = - \frac{9 - 9 i \sqrt{3}}{2}$ for $x$ .

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Step 25.2.3.5.1

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

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

Step 25.2.3.5.2

Simplify the exponent.

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Step 25.2.3.5.2.1

Simplify the left side.

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Step 25.2.3.5.2.1.1

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

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Step 25.2.3.5.2.1.1.1

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

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Step 25.2.3.5.2.1.1.1.1

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

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

Step 25.2.3.5.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 25.2.3.5.2.1.1.1.2.1

Cancel the common factor.

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

Step 25.2.3.5.2.1.1.1.2.2

Rewrite the expression.

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

Step 25.2.3.5.2.1.1.2

Simplify.

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

Step 25.2.3.5.2.2

Simplify the right side.

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Step 25.2.3.5.2.2.1

Simplify ${(- \frac{9 - 9 i \sqrt{3}}{2})}^{2}$ .

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Step 25.2.3.5.2.2.1.1

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

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Step 25.2.3.5.2.2.1.1.1

Apply the product rule to $- \frac{9 - 9 i \sqrt{3}}{2}$ .

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

Step 25.2.3.5.2.2.1.1.2

Apply the product rule to $\frac{9 - 9 i \sqrt{3}}{2}$ .

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

Step 25.2.3.5.2.2.1.2

Simplify the expression.

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Step 25.2.3.5.2.2.1.2.1

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

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

Step 25.2.3.5.2.2.1.2.2

Multiply $\frac{{(9 - 9 i \sqrt{3})}^{2}}{2^{2}}$ by $1$ .

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

Step 25.2.3.5.2.2.1.2.3

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

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

Step 25.2.3.5.2.2.1.2.4

Rewrite ${(9 - 9 i \sqrt{3})}^{2}$ as $(9 - 9 i \sqrt{3}) (9 - 9 i \sqrt{3})$ .

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

Step 25.2.3.5.2.2.1.3

Expand $(9 - 9 i \sqrt{3}) (9 - 9 i \sqrt{3})$ using the FOIL Method.

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Step 25.2.3.5.2.2.1.3.1

Apply the distributive property.

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

Step 25.2.3.5.2.2.1.3.2

Apply the distributive property.

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

Step 25.2.3.5.2.2.1.3.3

Apply the distributive property.

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

Step 25.2.3.5.2.2.1.4

Simplify and combine like terms.

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Step 25.2.3.5.2.2.1.4.1

Simplify each term.

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Step 25.2.3.5.2.2.1.4.1.1

Multiply $9$ by $9$ .

$x = \frac{81 + 9 (- 9 i \sqrt{3}) - 9 i \sqrt{3} \cdot 9 - 9 i \sqrt{3} (- 9 i \sqrt{3})}{4}$

Step 25.2.3.5.2.2.1.4.1.2

Multiply $- 9$ by $9$ .

$x = \frac{81 - 81 (i \sqrt{3}) - 9 i \sqrt{3} \cdot 9 - 9 i \sqrt{3} (- 9 i \sqrt{3})}{4}$

Step 25.2.3.5.2.2.1.4.1.3

Multiply $9$ by $- 9$ .

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} - 9 i \sqrt{3} (- 9 i \sqrt{3})}{4}$

Step 25.2.3.5.2.2.1.4.1.4

Multiply $- 9 i \sqrt{3} (- 9 i \sqrt{3})$ .

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Step 25.2.3.5.2.2.1.4.1.4.1

Multiply $- 9$ by $- 9$ .

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} + 81 i \sqrt{3} (i \sqrt{3})}{4}$

Step 25.2.3.5.2.2.1.4.1.4.2

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

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} + 81 (i^{1} i) \sqrt{3} \sqrt{3}}{4}$

Step 25.2.3.5.2.2.1.4.1.4.3

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

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} + 81 (i^{1} i^{1}) \sqrt{3} \sqrt{3}}{4}$

Step 25.2.3.5.2.2.1.4.1.4.4

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

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} + 81 i^{1 + 1} \sqrt{3} \sqrt{3}}{4}$

Step 25.2.3.5.2.2.1.4.1.4.5

Add $1$ and $1$ .

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} + 81 i^{2} \sqrt{3} \sqrt{3}}{4}$

Step 25.2.3.5.2.2.1.4.1.4.6

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

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} + 81 i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{4}$

Step 25.2.3.5.2.2.1.4.1.4.7

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

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} + 81 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4}$

Step 25.2.3.5.2.2.1.4.1.4.8

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

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} + 81 i^{2} {\sqrt{3}}^{1 + 1}}{4}$

Step 25.2.3.5.2.2.1.4.1.4.9

Add $1$ and $1$ .

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} + 81 i^{2} {\sqrt{3}}^{2}}{4}$

Step 25.2.3.5.2.2.1.4.1.5

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

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} + 81 \cdot - 1 {\sqrt{3}}^{2}}{4}$

Step 25.2.3.5.2.2.1.4.1.6

Multiply $81$ by $- 1$ .

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} - 81 {\sqrt{3}}^{2}}{4}$

Step 25.2.3.5.2.2.1.4.1.7

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

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Step 25.2.3.5.2.2.1.4.1.7.1

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

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} - 81 {(3^{\frac{1}{2}})}^{2}}{4}$

Step 25.2.3.5.2.2.1.4.1.7.2

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

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} - 81 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 25.2.3.5.2.2.1.4.1.7.3

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

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} - 81 \cdot 3^{\frac{2}{2}}}{4}$

Step 25.2.3.5.2.2.1.4.1.7.4

Cancel the common factor of $2$ .

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Step 25.2.3.5.2.2.1.4.1.7.4.1

Cancel the common factor.

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} - 81 \cdot 3^{\frac{2}{2}}}{4}$

Step 25.2.3.5.2.2.1.4.1.7.4.2

Rewrite the expression.

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} - 81 \cdot 3^{1}}{4}$

Step 25.2.3.5.2.2.1.4.1.7.5

Evaluate the exponent.

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} - 81 \cdot 3}{4}$

Step 25.2.3.5.2.2.1.4.1.8

Multiply $- 81$ by $3$ .

$x = \frac{81 - 81 i \sqrt{3} - 81 i \sqrt{3} - 243}{4}$

Step 25.2.3.5.2.2.1.4.2

Subtract $243$ from $81$ .

$x = \frac{- 81 i \sqrt{3} - 81 i \sqrt{3} - 162}{4}$

Step 25.2.3.5.2.2.1.4.3

Subtract $81 i \sqrt{3}$ from $- 81 i \sqrt{3}$ .

$x = \frac{- 162 i \sqrt{3} - 162}{4}$

Step 25.2.3.5.2.2.1.5

Reorder $- 162 i \sqrt{3}$ and $- 162$ .

$x = \frac{- 162 - 162 i \sqrt{3}}{4}$

Step 25.2.3.5.2.2.1.6

Cancel the common factor of $- 162 - 162 i \sqrt{3}$ and $4$ .

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Step 25.2.3.5.2.2.1.6.1

Factor $2$ out of $- 162$ .

$x = \frac{2 (- 81) - 162 i \sqrt{3}}{4}$

Step 25.2.3.5.2.2.1.6.2

Factor $2$ out of $- 162 i \sqrt{3}$ .

$x = \frac{2 (- 81) + 2 (- 81 i \sqrt{3})}{4}$

Step 25.2.3.5.2.2.1.6.3

Factor $2$ out of $2 (- 81) + 2 (- 81 i \sqrt{3})$ .

$x = \frac{2 (- 81 - 81 i \sqrt{3})}{4}$

Step 25.2.3.5.2.2.1.6.4

Cancel the common factors.

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Step 25.2.3.5.2.2.1.6.4.1

Factor $2$ out of $4$ .

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

Step 25.2.3.5.2.2.1.6.4.2

Cancel the common factor.

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

Step 25.2.3.5.2.2.1.6.4.3

Rewrite the expression.

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

Step 25.2.3.5.2.2.1.7

Rewrite $- 81$ as $- 1 (81)$ .

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

Step 25.2.3.5.2.2.1.8

Factor $- 1$ out of $- 81 i \sqrt{3}$ .

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

Step 25.2.3.5.2.2.1.9

Factor $- 1$ out of $- 1 (81) - (81 i \sqrt{3})$ .

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

Step 25.2.3.5.2.2.1.10

Move the negative in front of the fraction.

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

Step 25.2.3.6

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

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Step 25.2.3.6.1

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

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

Step 25.2.3.6.2

Simplify the exponent.

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Step 25.2.3.6.2.1

Simplify the left side.

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Step 25.2.3.6.2.1.1

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

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Step 25.2.3.6.2.1.1.1

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

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Step 25.2.3.6.2.1.1.1.1

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

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

Step 25.2.3.6.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 25.2.3.6.2.1.1.1.2.1

Cancel the common factor.

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

Step 25.2.3.6.2.1.1.1.2.2

Rewrite the expression.

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

Step 25.2.3.6.2.1.1.2

Simplify.

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

Step 25.2.3.6.2.2

Simplify the right side.

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Step 25.2.3.6.2.2.1

Simplify ${(- \frac{9 + 9 i \sqrt{3}}{2})}^{2}$ .

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Step 25.2.3.6.2.2.1.1

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

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Step 25.2.3.6.2.2.1.1.1

Apply the product rule to $- \frac{9 + 9 i \sqrt{3}}{2}$ .

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

Step 25.2.3.6.2.2.1.1.2

Apply the product rule to $\frac{9 + 9 i \sqrt{3}}{2}$ .

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

Step 25.2.3.6.2.2.1.2

Simplify the expression.

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Step 25.2.3.6.2.2.1.2.1

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

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

Step 25.2.3.6.2.2.1.2.2

Multiply $\frac{{(9 + 9 i \sqrt{3})}^{2}}{2^{2}}$ by $1$ .

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

Step 25.2.3.6.2.2.1.2.3

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

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

Step 25.2.3.6.2.2.1.2.4

Rewrite ${(9 + 9 i \sqrt{3})}^{2}$ as $(9 + 9 i \sqrt{3}) (9 + 9 i \sqrt{3})$ .

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

Step 25.2.3.6.2.2.1.3

Expand $(9 + 9 i \sqrt{3}) (9 + 9 i \sqrt{3})$ using the FOIL Method.

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Step 25.2.3.6.2.2.1.3.1

Apply the distributive property.

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

Step 25.2.3.6.2.2.1.3.2

Apply the distributive property.

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

Step 25.2.3.6.2.2.1.3.3

Apply the distributive property.

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

Step 25.2.3.6.2.2.1.4

Simplify and combine like terms.

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Step 25.2.3.6.2.2.1.4.1

Simplify each term.

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Step 25.2.3.6.2.2.1.4.1.1

Multiply $9$ by $9$ .

$x = \frac{81 + 9 (9 i \sqrt{3}) + 9 i \sqrt{3} \cdot 9 + 9 i \sqrt{3} (9 i \sqrt{3})}{4}$

Step 25.2.3.6.2.2.1.4.1.2

Multiply $9$ by $9$ .

$x = \frac{81 + 81 (i \sqrt{3}) + 9 i \sqrt{3} \cdot 9 + 9 i \sqrt{3} (9 i \sqrt{3})}{4}$

Step 25.2.3.6.2.2.1.4.1.3

Multiply $9$ by $9$ .

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} + 9 i \sqrt{3} (9 i \sqrt{3})}{4}$

Step 25.2.3.6.2.2.1.4.1.4

Multiply $9 i \sqrt{3} (9 i \sqrt{3})$ .

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Step 25.2.3.6.2.2.1.4.1.4.1

Multiply $9$ by $9$ .

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} + 81 i \sqrt{3} (i \sqrt{3})}{4}$

Step 25.2.3.6.2.2.1.4.1.4.2

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

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} + 81 (i^{1} i) \sqrt{3} \sqrt{3}}{4}$

Step 25.2.3.6.2.2.1.4.1.4.3

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

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} + 81 (i^{1} i^{1}) \sqrt{3} \sqrt{3}}{4}$

Step 25.2.3.6.2.2.1.4.1.4.4

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

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} + 81 i^{1 + 1} \sqrt{3} \sqrt{3}}{4}$

Step 25.2.3.6.2.2.1.4.1.4.5

Add $1$ and $1$ .

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} + 81 i^{2} \sqrt{3} \sqrt{3}}{4}$

Step 25.2.3.6.2.2.1.4.1.4.6

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

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} + 81 i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{4}$

Step 25.2.3.6.2.2.1.4.1.4.7

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

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} + 81 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4}$

Step 25.2.3.6.2.2.1.4.1.4.8

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

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} + 81 i^{2} {\sqrt{3}}^{1 + 1}}{4}$

Step 25.2.3.6.2.2.1.4.1.4.9

Add $1$ and $1$ .

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} + 81 i^{2} {\sqrt{3}}^{2}}{4}$

Step 25.2.3.6.2.2.1.4.1.5

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

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} + 81 \cdot - 1 {\sqrt{3}}^{2}}{4}$

Step 25.2.3.6.2.2.1.4.1.6

Multiply $81$ by $- 1$ .

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} - 81 {\sqrt{3}}^{2}}{4}$

Step 25.2.3.6.2.2.1.4.1.7

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

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Step 25.2.3.6.2.2.1.4.1.7.1

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

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} - 81 {(3^{\frac{1}{2}})}^{2}}{4}$

Step 25.2.3.6.2.2.1.4.1.7.2

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

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} - 81 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 25.2.3.6.2.2.1.4.1.7.3

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

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} - 81 \cdot 3^{\frac{2}{2}}}{4}$

Step 25.2.3.6.2.2.1.4.1.7.4

Cancel the common factor of $2$ .

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Step 25.2.3.6.2.2.1.4.1.7.4.1

Cancel the common factor.

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} - 81 \cdot 3^{\frac{2}{2}}}{4}$

Step 25.2.3.6.2.2.1.4.1.7.4.2

Rewrite the expression.

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} - 81 \cdot 3^{1}}{4}$

Step 25.2.3.6.2.2.1.4.1.7.5

Evaluate the exponent.

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} - 81 \cdot 3}{4}$

Step 25.2.3.6.2.2.1.4.1.8

Multiply $- 81$ by $3$ .

$x = \frac{81 + 81 i \sqrt{3} + 81 i \sqrt{3} - 243}{4}$

Step 25.2.3.6.2.2.1.4.2

Subtract $243$ from $81$ .

$x = \frac{81 i \sqrt{3} + 81 i \sqrt{3} - 162}{4}$

Step 25.2.3.6.2.2.1.4.3

Add $81 i \sqrt{3}$ and $81 i \sqrt{3}$ .

$x = \frac{162 i \sqrt{3} - 162}{4}$

Step 25.2.3.6.2.2.1.5

Reorder $162 i \sqrt{3}$ and $- 162$ .

$x = \frac{- 162 + 162 i \sqrt{3}}{4}$

Step 25.2.3.6.2.2.1.6

Cancel the common factor of $- 162 + 162 i \sqrt{3}$ and $4$ .

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Step 25.2.3.6.2.2.1.6.1

Factor $2$ out of $- 162$ .

$x = \frac{2 (- 81) + 162 i \sqrt{3}}{4}$

Step 25.2.3.6.2.2.1.6.2

Factor $2$ out of $162 i \sqrt{3}$ .

$x = \frac{2 (- 81) + 2 (81 i \sqrt{3})}{4}$

Step 25.2.3.6.2.2.1.6.3

Factor $2$ out of $2 (- 81) + 2 (81 i \sqrt{3})$ .

$x = \frac{2 (- 81 + 81 i \sqrt{3})}{4}$

Step 25.2.3.6.2.2.1.6.4

Cancel the common factors.

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Step 25.2.3.6.2.2.1.6.4.1

Factor $2$ out of $4$ .

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

Step 25.2.3.6.2.2.1.6.4.2

Cancel the common factor.

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

Step 25.2.3.6.2.2.1.6.4.3

Rewrite the expression.

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

Step 25.2.3.6.2.2.1.7

Rewrite $- 81$ as $- 1 (81)$ .

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

Step 25.2.3.6.2.2.1.8

Factor $- 1$ out of $81 i \sqrt{3}$ .

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

Step 25.2.3.6.2.2.1.9

Factor $- 1$ out of $- 1 (81) - (- 81 i \sqrt{3})$ .

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

Step 25.2.3.6.2.2.1.10

Move the negative in front of the fraction.

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

Step 25.2.3.7

List all of the solutions.

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

Step 26

The final solution is all the values that make $\frac{9 - x^{\frac{1}{2}}}{9 x^{\frac{1}{2}}} \cdot \frac{81 + 9 x^{\frac{1}{2}} + x}{81 x} = 0$ true.

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