Algebra Examples

${(x^{\frac{3}{2}})}^{2} = \sqrt[4]{x^{3}}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt[4]{x^{3}} = {(x^{\frac{3}{2}})}^{2}$

Step 2

Subtract ${(x^{\frac{3}{2}})}^{2}$ from both sides of the equation.

$\sqrt[4]{x^{3}} - {(x^{\frac{3}{2}})}^{2} = 0$

Step 3

Multiply the exponents in ${(x^{\frac{3}{2}})}^{2}$ .

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Step 3.1

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

$\sqrt[4]{x^{3}} - x^{\frac{3}{2} \cdot 2} = 0$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

$\sqrt[4]{x^{3}} - x^{\frac{3}{2} \cdot 2} = 0$

Step 3.2.2

Rewrite the expression.

$\sqrt[4]{x^{3}} - x^{3} = 0$

Step 4

Factor $\sqrt[4]{x^{3}} - x^{3}$ .

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

$x^{\frac{3}{4}} - x^{3} = 0$

Step 4.2

Factor $x^{\frac{3}{4}}$ out of $x^{\frac{3}{4}} - x^{3}$ .

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Step 4.2.1

Multiply by $1$ .

$x^{\frac{3}{4}} \cdot 1 - x^{3} = 0$

Step 4.2.2

Factor $x^{\frac{3}{4}}$ out of $- x^{3}$ .

$x^{\frac{3}{4}} \cdot 1 + x^{\frac{3}{4}} (- x^{\frac{9}{4}}) = 0$

Step 4.2.3

Factor $x^{\frac{3}{4}}$ out of $x^{\frac{3}{4}} \cdot 1 + x^{\frac{3}{4}} (- x^{\frac{9}{4}})$ .

$x^{\frac{3}{4}} (1 - x^{\frac{9}{4}}) = 0$

Step 4.3

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

$x^{\frac{3}{4}} (1^{3} - x^{\frac{9}{4}}) = 0$

Step 4.4

Rewrite $x^{\frac{9}{4}}$ as ${(x^{\frac{3}{4}})}^{3}$ .

$x^{\frac{3}{4}} (1^{3} - {(x^{\frac{3}{4}})}^{3}) = 0$

Step 4.5

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 $a = 1$ and $b = x^{\frac{3}{4}}$ .

$x^{\frac{3}{4}} ((1 - x^{\frac{3}{4}}) (1^{2} + 1 x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6

Factor.

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Step 4.6.1

Simplify.

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Step 4.6.1.1

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

$x^{\frac{3}{4}} ((1^{3} - x^{\frac{3}{4}}) (1^{2} + 1 x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6.1.2

Rewrite $x^{\frac{3}{4}}$ as ${(x^{\frac{1}{4}})}^{3}$ .

$x^{\frac{3}{4}} ((1^{3} - {(x^{\frac{1}{4}})}^{3}) (1^{2} + 1 x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6.1.3

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 $a = 1$ and $b = x^{\frac{1}{4}}$ .

$x^{\frac{3}{4}} ((1 - x^{\frac{1}{4}}) (1^{2} + 1 x^{\frac{1}{4}} + {(x^{\frac{1}{4}})}^{2}) (1^{2} + 1 x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6.1.4

Simplify.

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Step 4.6.1.4.1

One to any power is one.

$x^{\frac{3}{4}} ((1 - x^{\frac{1}{4}}) (1 + 1 x^{\frac{1}{4}} + {(x^{\frac{1}{4}})}^{2}) (1^{2} + 1 x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6.1.4.2

Multiply $x^{\frac{1}{4}}$ by $1$ .

$x^{\frac{3}{4}} ((1 - x^{\frac{1}{4}}) (1 + x^{\frac{1}{4}} + {(x^{\frac{1}{4}})}^{2}) (1^{2} + 1 x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6.1.4.3

Multiply the exponents in ${(x^{\frac{1}{4}})}^{2}$ .

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Step 4.6.1.4.3.1

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

$x^{\frac{3}{4}} ((1 - x^{\frac{1}{4}}) (1 + x^{\frac{1}{4}} + x^{\frac{1}{4} \cdot 2}) (1^{2} + 1 x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6.1.4.3.2

Cancel the common factor of $2$ .

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Step 4.6.1.4.3.2.1

Factor $2$ out of $4$ .

$x^{\frac{3}{4}} ((1 - x^{\frac{1}{4}}) (1 + x^{\frac{1}{4}} + x^{\frac{1}{2 (2)} \cdot 2}) (1^{2} + 1 x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6.1.4.3.2.2

Cancel the common factor.

$x^{\frac{3}{4}} ((1 - x^{\frac{1}{4}}) (1 + x^{\frac{1}{4}} + x^{\frac{1}{2 \cdot 2} \cdot 2}) (1^{2} + 1 x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6.1.4.3.2.3

Rewrite the expression.

$x^{\frac{3}{4}} ((1 - x^{\frac{1}{4}}) (1 + x^{\frac{1}{4}} + x^{\frac{1}{2}}) (1^{2} + 1 x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6.1.4.4

Reorder terms.

$x^{\frac{3}{4}} ((1 - x^{\frac{1}{4}}) (x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1) (1^{2} + 1 x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6.1.5

Multiply $x^{\frac{3}{4}}$ by $1$ .

$x^{\frac{3}{4}} ((1 - x^{\frac{1}{4}}) (x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1) (1^{2} + x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2})) = 0$

Step 4.6.2

Remove unnecessary parentheses.

$x^{\frac{3}{4}} (1 - x^{\frac{1}{4}}) (x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1) (1^{2} + x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2}) = 0$

Step 4.7

Simplify each term.

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Step 4.7.1

One to any power is one.

$x^{\frac{3}{4}} (1 - x^{\frac{1}{4}}) (x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1) (1 + x^{\frac{3}{4}} + {(x^{\frac{3}{4}})}^{2}) = 0$

Step 4.7.2

Multiply the exponents in ${(x^{\frac{3}{4}})}^{2}$ .

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Step 4.7.2.1

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

$x^{\frac{3}{4}} (1 - x^{\frac{1}{4}}) (x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1) (1 + x^{\frac{3}{4}} + x^{\frac{3}{4} \cdot 2}) = 0$

Step 4.7.2.2

Cancel the common factor of $2$ .

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Step 4.7.2.2.1

Factor $2$ out of $4$ .

$x^{\frac{3}{4}} (1 - x^{\frac{1}{4}}) (x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1) (1 + x^{\frac{3}{4}} + x^{\frac{3}{2 (2)} \cdot 2}) = 0$

Step 4.7.2.2.2

Cancel the common factor.

$x^{\frac{3}{4}} (1 - x^{\frac{1}{4}}) (x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1) (1 + x^{\frac{3}{4}} + x^{\frac{3}{2 \cdot 2} \cdot 2}) = 0$

Step 4.7.2.2.3

Rewrite the expression.

$x^{\frac{3}{4}} (1 - x^{\frac{1}{4}}) (x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1) (1 + x^{\frac{3}{4}} + x^{\frac{3}{2}}) = 0$

Step 4.8

Reorder terms.

$x^{\frac{3}{4}} (1 - x^{\frac{1}{4}}) (x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1) (x^{\frac{3}{4}} + x^{\frac{3}{2}} + 1) = 0$

Step 5

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

$x^{\frac{3}{4}} = 0$

$1 - x^{\frac{1}{4}} = 0$

$x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1 = 0$

$x^{\frac{3}{4}} + x^{\frac{3}{2}} + 1 = 0$

Step 6

Set $x^{\frac{3}{4}}$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{\frac{3}{4}}$ equal to $0$ .

$x^{\frac{3}{4}} = 0$

Step 6.2

Solve $x^{\frac{3}{4}} = 0$ for $x$ .

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Step 6.2.1

Raise each side of the equation to the power of $\frac{4}{3}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{3}{4}})}^{\frac{4}{3}} = 0^{\frac{4}{3}}$

Step 6.2.2

Simplify the exponent.

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Step 6.2.2.1

Simplify the left side.

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Step 6.2.2.1.1

Simplify ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 6.2.2.1.1.1

Multiply the exponents in ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 6.2.2.1.1.1.1

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

$x^{\frac{3}{4} \cdot \frac{4}{3}} = 0^{\frac{4}{3}}$

Step 6.2.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 6.2.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{3}{4} \cdot \frac{4}{3}} = 0^{\frac{4}{3}}$

Step 6.2.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{4} \cdot 4} = 0^{\frac{4}{3}}$

Step 6.2.2.1.1.1.3

Cancel the common factor of $4$ .

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Step 6.2.2.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = 0^{\frac{4}{3}}$

Step 6.2.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = 0^{\frac{4}{3}}$

Step 6.2.2.1.1.2

Simplify.

$x = 0^{\frac{4}{3}}$

Step 6.2.2.2

Simplify the right side.

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Step 6.2.2.2.1

Simplify $0^{\frac{4}{3}}$ .

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Step 6.2.2.2.1.1

Simplify the expression.

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Step 6.2.2.2.1.1.1

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

$x = {(0^{3})}^{\frac{4}{3}}$

Step 6.2.2.2.1.1.2

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

$x = 0^{3 (\frac{4}{3})}$

Step 6.2.2.2.1.2

Cancel the common factor of $3$ .

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Step 6.2.2.2.1.2.1

Cancel the common factor.

$x = 0^{3 (\frac{4}{3})}$

Step 6.2.2.2.1.2.2

Rewrite the expression.

$x = 0^{4}$

Step 6.2.2.2.1.3

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 7

Set $1 - x^{\frac{1}{4}}$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $1 - x^{\frac{1}{4}}$ equal to $0$ .

$1 - x^{\frac{1}{4}} = 0$

Step 7.2

Solve $1 - x^{\frac{1}{4}} = 0$ for $x$ .

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Step 7.2.1

Subtract $1$ from both sides of the equation.

$- x^{\frac{1}{4}} = - 1$

Step 7.2.2

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${(- x^{\frac{1}{4}})}^{4} = {(- 1)}^{4}$

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 ${(- x^{\frac{1}{4}})}^{4}$ .

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Step 7.2.3.1.1.1

Apply the product rule to $- x^{\frac{1}{4}}$ .

${(- 1)}^{4} {(x^{\frac{1}{4}})}^{4} = {(- 1)}^{4}$

Step 7.2.3.1.1.2

Raise $- 1$ to the power of $4$ .

$1 {(x^{\frac{1}{4}})}^{4} = {(- 1)}^{4}$

Step 7.2.3.1.1.3

Multiply ${(x^{\frac{1}{4}})}^{4}$ by $1$ .

${(x^{\frac{1}{4}})}^{4} = {(- 1)}^{4}$

Step 7.2.3.1.1.4

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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Step 7.2.3.1.1.4.1

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

$x^{\frac{1}{4} \cdot 4} = {(- 1)}^{4}$

Step 7.2.3.1.1.4.2

Cancel the common factor of $4$ .

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Step 7.2.3.1.1.4.2.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = {(- 1)}^{4}$

Step 7.2.3.1.1.4.2.2

Rewrite the expression.

$x^{1} = {(- 1)}^{4}$

Step 7.2.3.1.1.5

Simplify.

$x = {(- 1)}^{4}$

Step 7.2.3.2

Simplify the right side.

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Step 7.2.3.2.1

Raise $- 1$ to the power of $4$ .

$x = 1$

Step 8

Set $x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1$ equal to $0$ .

$x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1 = 0$

Step 8.2

Solve $x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1 = 0$ for $x$ .

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Step 8.2.1

Find a common factor $x^{\frac{1}{4}}$ that is present in each term.

$x^{\frac{1}{4}} {(x^{\frac{1}{4}})}^{2} + 1$

Step 8.2.2

Substitute $u$ for $x^{\frac{1}{4}}$ .

$(u) {(u)}^{2} + 1 = 0$

Step 8.2.3

Solve for $u$ .

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Step 8.2.3.1

Multiply $u$ by ${(u)}^{2}$ by adding the exponents.

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Step 8.2.3.1.1

Multiply $u$ by ${(u)}^{2}$ .

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Step 8.2.3.1.1.1

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

$u^{1} {(u)}^{2} + 1 = 0$

Step 8.2.3.1.1.2

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

$u^{1 + 2} + 1 = 0$

Step 8.2.3.1.2

Add $1$ and $2$ .

$u^{3} + 1 = 0$

Step 8.2.3.2

Subtract $1$ from both sides of the equation.

$u^{3} = - 1$

Step 8.2.3.3

Add $1$ to both sides of the equation.

$u^{3} + 1 = 0$

Step 8.2.3.4

Factor the left side of the equation.

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Step 8.2.3.4.1

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

$u^{3} + 1^{3} = 0$

Step 8.2.3.4.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = u$ and $b = 1$ .

$(u + 1) (u^{2} - u \cdot 1 + 1^{2}) = 0$

Step 8.2.3.4.3

Simplify.

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Step 8.2.3.4.3.1

Multiply $- 1$ by $1$ .

$(u + 1) (u^{2} - u + 1^{2}) = 0$

Step 8.2.3.4.3.2

One to any power is one.

$(u + 1) (u^{2} - u + 1) = 0$

Step 8.2.3.5

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

$u + 1 = 0$

$u^{2} - u + 1 = 0$

Step 8.2.3.6

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 8.2.3.6.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 8.2.3.6.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 8.2.3.7

Set $u^{2} - u + 1$ equal to $0$ and solve for $u$ .

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Step 8.2.3.7.1

Set $u^{2} - u + 1$ equal to $0$ .

$u^{2} - u + 1 = 0$

Step 8.2.3.7.2

Solve $u^{2} - u + 1 = 0$ for $u$ .

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Step 8.2.3.7.2.1

Use the quadratic formula to find the solutions.

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

Step 8.2.3.7.2.2

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

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

Step 8.2.3.7.2.3

Simplify.

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Step 8.2.3.7.2.3.1

Simplify the numerator.

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Step 8.2.3.7.2.3.1.1

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

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

Step 8.2.3.7.2.3.1.2

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

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Step 8.2.3.7.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 8.2.3.7.2.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 8.2.3.7.2.3.1.3

Subtract $4$ from $1$ .

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

Step 8.2.3.7.2.3.1.4

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

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

Step 8.2.3.7.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 8.2.3.7.2.3.1.6

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

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

Step 8.2.3.7.2.3.2

Multiply $2$ by $1$ .

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

Step 8.2.3.7.2.4

The final answer is the combination of both solutions.

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

Step 8.2.3.8

The final solution is all the values that make $(u + 1) (u^{2} - u + 1) = 0$ true.

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

Step 8.2.4

Substitute $x$ for $u$ .

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

Step 8.2.5

Solve for $x^{\frac{1}{4}} = - 1$ for $x$ .

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Step 8.2.5.1

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{4}})}^{4} = {(- 1)}^{4}$

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 ${(x^{\frac{1}{4}})}^{4}$ .

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Step 8.2.5.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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Step 8.2.5.2.1.1.1.1

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

$x^{\frac{1}{4} \cdot 4} = {(- 1)}^{4}$

Step 8.2.5.2.1.1.1.2

Cancel the common factor of $4$ .

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Step 8.2.5.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = {(- 1)}^{4}$

Step 8.2.5.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(- 1)}^{4}$

Step 8.2.5.2.1.1.2

Simplify.

$x = {(- 1)}^{4}$

Step 8.2.5.2.2

Simplify the right side.

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Step 8.2.5.2.2.1

Raise $- 1$ to the power of $4$ .

$x = 1$

Step 8.2.6

Solve for $x^{\frac{1}{4}} = \frac{1 + i \sqrt{3}}{2}$ for $x$ .

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Step 8.2.6.1

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

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

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 ${(x^{\frac{1}{4}})}^{4}$ .

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Step 8.2.6.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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Step 8.2.6.2.1.1.1.1

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

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

Step 8.2.6.2.1.1.1.2

Cancel the common factor of $4$ .

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Step 8.2.6.2.1.1.1.2.1

Cancel the common factor.

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

Step 8.2.6.2.1.1.1.2.2

Rewrite the expression.

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

Step 8.2.6.2.1.1.2

Simplify.

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

Step 8.2.6.2.2

Simplify the right side.

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Step 8.2.6.2.2.1

Simplify ${(\frac{1 + i \sqrt{3}}{2})}^{4}$ .

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Step 8.2.6.2.2.1.1

Simplify the expression.

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Step 8.2.6.2.2.1.1.1

Apply the product rule to $\frac{1 + i \sqrt{3}}{2}$ .

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

Step 8.2.6.2.2.1.1.2

Raise $2$ to the power of $4$ .

$x = \frac{{(1 + i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.2

Use the Binomial Theorem.

$x = \frac{1^{4} + 4 \cdot 1^{3} (i \sqrt{3}) + 6 \cdot 1^{2} {(i \sqrt{3})}^{2} + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3

Simplify terms.

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Step 8.2.6.2.2.1.3.1

Simplify each term.

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Step 8.2.6.2.2.1.3.1.1

One to any power is one.

$x = \frac{1 + 4 \cdot 1^{3} (i \sqrt{3}) + 6 \cdot 1^{2} {(i \sqrt{3})}^{2} + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.2

One to any power is one.

$x = \frac{1 + 4 \cdot 1 (i \sqrt{3}) + 6 \cdot 1^{2} {(i \sqrt{3})}^{2} + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.3

Multiply $4$ by $1$ .

$x = \frac{1 + 4 (i \sqrt{3}) + 6 \cdot 1^{2} {(i \sqrt{3})}^{2} + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.4

One to any power is one.

$x = \frac{1 + 4 i \sqrt{3} + 6 \cdot 1 {(i \sqrt{3})}^{2} + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.5

Multiply $6$ by $1$ .

$x = \frac{1 + 4 i \sqrt{3} + 6 {(i \sqrt{3})}^{2} + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.6

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

$x = \frac{1 + 4 i \sqrt{3} + 6 (i^{2} {\sqrt{3}}^{2}) + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.7

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

$x = \frac{1 + 4 i \sqrt{3} + 6 (- 1 {\sqrt{3}}^{2}) + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.8

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

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Step 8.2.6.2.2.1.3.1.8.1

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

$x = \frac{1 + 4 i \sqrt{3} + 6 (- 1 {(3^{\frac{1}{2}})}^{2}) + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.8.2

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

$x = \frac{1 + 4 i \sqrt{3} + 6 (- 1 \cdot 3^{\frac{1}{2} \cdot 2}) + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.8.3

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

$x = \frac{1 + 4 i \sqrt{3} + 6 (- 1 \cdot 3^{\frac{2}{2}}) + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.8.4

Cancel the common factor of $2$ .

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Step 8.2.6.2.2.1.3.1.8.4.1

Cancel the common factor.

$x = \frac{1 + 4 i \sqrt{3} + 6 (- 1 \cdot 3^{\frac{2}{2}}) + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.8.4.2

Rewrite the expression.

$x = \frac{1 + 4 i \sqrt{3} + 6 (- 1 \cdot 3^{1}) + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.8.5

Evaluate the exponent.

$x = \frac{1 + 4 i \sqrt{3} + 6 (- 1 \cdot 3) + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.9

Multiply $6 (- 1 \cdot 3)$ .

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Step 8.2.6.2.2.1.3.1.9.1

Multiply $- 1$ by $3$ .

$x = \frac{1 + 4 i \sqrt{3} + 6 \cdot - 3 + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.9.2

Multiply $6$ by $- 3$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 \cdot 1 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.10

Multiply $4$ by $1$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 {(i \sqrt{3})}^{3} + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.11

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

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 (i^{3} {\sqrt{3}}^{3}) + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.12

Factor out $i^{2}$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 (i^{2} \cdot i {\sqrt{3}}^{3}) + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.13

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

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 (- 1 \cdot i {\sqrt{3}}^{3}) + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.14

Rewrite $- 1 i$ as $- i$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 (- i {\sqrt{3}}^{3}) + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.15

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

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 (- i \sqrt{3^{3}}) + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.16

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

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 (- i \sqrt{27}) + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.17

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

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Step 8.2.6.2.2.1.3.1.17.1

Factor $9$ out of $27$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 (- i \sqrt{9 (3)}) + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.17.2

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

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 (- i \sqrt{3^{2} \cdot 3}) + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.18

Pull terms out from under the radical.

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 (- i (3 \sqrt{3})) + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.19

Multiply $3$ by $- 1$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 + 4 (- 3 i \sqrt{3}) + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.20

Multiply $- 3$ by $4$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 (i \sqrt{3}) + {(i \sqrt{3})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.21

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

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + i^{4} {\sqrt{3}}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.22

Rewrite $i^{4}$ as $1$ .

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Step 8.2.6.2.2.1.3.1.22.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + {(i^{2})}^{2} {\sqrt{3}}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.22.2

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

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + {(- 1)}^{2} {\sqrt{3}}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.22.3

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

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 1 {\sqrt{3}}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.23

Multiply ${\sqrt{3}}^{4}$ by $1$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + {\sqrt{3}}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.24

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 8.2.6.2.2.1.3.1.24.1

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

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + {(3^{\frac{1}{2}})}^{4}}{16}$

Step 8.2.6.2.2.1.3.1.24.2

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

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{\frac{1}{2} \cdot 4}}{16}$

Step 8.2.6.2.2.1.3.1.24.3

Combine $\frac{1}{2}$ and $4$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{\frac{4}{2}}}{16}$

Step 8.2.6.2.2.1.3.1.24.4

Cancel the common factor of $4$ and $2$ .

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Step 8.2.6.2.2.1.3.1.24.4.1

Factor $2$ out of $4$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{\frac{2 \cdot 2}{2}}}{16}$

Step 8.2.6.2.2.1.3.1.24.4.2

Cancel the common factors.

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Step 8.2.6.2.2.1.3.1.24.4.2.1

Factor $2$ out of $2$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{\frac{2 \cdot 2}{2 (1)}}}{16}$

Step 8.2.6.2.2.1.3.1.24.4.2.2

Cancel the common factor.

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{\frac{2 \cdot 2}{2 \cdot 1}}}{16}$

Step 8.2.6.2.2.1.3.1.24.4.2.3

Rewrite the expression.

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{\frac{2}{1}}}{16}$

Step 8.2.6.2.2.1.3.1.24.4.2.4

Divide $2$ by $1$ .

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{2}}{16}$

Step 8.2.6.2.2.1.3.1.25

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

$x = \frac{1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 9}{16}$

Step 8.2.6.2.2.1.3.2

Simplify terms.

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Step 8.2.6.2.2.1.3.2.1

Subtract $18$ from $1$ .

$x = \frac{4 i \sqrt{3} - 17 - 12 i \sqrt{3} + 9}{16}$

Step 8.2.6.2.2.1.3.2.2

Subtract $12 i \sqrt{3}$ from $4 i \sqrt{3}$ .

$x = \frac{- 8 i \sqrt{3} - 17 + 9}{16}$

Step 8.2.6.2.2.1.3.2.3

Simplify the expression.

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Step 8.2.6.2.2.1.3.2.3.1

Add $- 17$ and $9$ .

$x = \frac{- 8 i \sqrt{3} - 8}{16}$

Step 8.2.6.2.2.1.3.2.3.2

Reorder $- 8 i \sqrt{3}$ and $- 8$ .

$x = \frac{- 8 - 8 i \sqrt{3}}{16}$

Step 8.2.6.2.2.1.3.2.4

Cancel the common factor of $- 8 - 8 i \sqrt{3}$ and $16$ .

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Step 8.2.6.2.2.1.3.2.4.1

Factor $8$ out of $- 8$ .

$x = \frac{8 (- 1) - 8 i \sqrt{3}}{16}$

Step 8.2.6.2.2.1.3.2.4.2

Factor $8$ out of $- 8 i \sqrt{3}$ .

$x = \frac{8 (- 1) + 8 (- i \sqrt{3})}{16}$

Step 8.2.6.2.2.1.3.2.4.3

Factor $8$ out of $8 (- 1) + 8 (- i \sqrt{3})$ .

$x = \frac{8 (- 1 - i \sqrt{3})}{16}$

Step 8.2.6.2.2.1.3.2.4.4

Cancel the common factors.

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Step 8.2.6.2.2.1.3.2.4.4.1

Factor $8$ out of $16$ .

$x = \frac{8 (- 1 - i \sqrt{3})}{8 \cdot 2}$

Step 8.2.6.2.2.1.3.2.4.4.2

Cancel the common factor.

$x = \frac{8 (- 1 - i \sqrt{3})}{8 \cdot 2}$

Step 8.2.6.2.2.1.3.2.4.4.3

Rewrite the expression.

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

Step 8.2.6.2.2.1.3.2.5

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

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

Step 8.2.6.2.2.1.3.2.6

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

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

Step 8.2.6.2.2.1.3.2.7

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

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

Step 8.2.6.2.2.1.3.2.8

Move the negative in front of the fraction.

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

Step 8.2.7

Solve for $x^{\frac{1}{4}} = \frac{1 - i \sqrt{3}}{2}$ for $x$ .

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Step 8.2.7.1

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

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

Step 8.2.7.2

Simplify the exponent.

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Step 8.2.7.2.1

Simplify the left side.

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Step 8.2.7.2.1.1

Simplify ${(x^{\frac{1}{4}})}^{4}$ .

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Step 8.2.7.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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Step 8.2.7.2.1.1.1.1

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

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

Step 8.2.7.2.1.1.1.2

Cancel the common factor of $4$ .

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Step 8.2.7.2.1.1.1.2.1

Cancel the common factor.

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

Step 8.2.7.2.1.1.1.2.2

Rewrite the expression.

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

Step 8.2.7.2.1.1.2

Simplify.

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

Step 8.2.7.2.2

Simplify the right side.

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Step 8.2.7.2.2.1

Simplify ${(\frac{1 - i \sqrt{3}}{2})}^{4}$ .

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Step 8.2.7.2.2.1.1

Simplify the expression.

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Step 8.2.7.2.2.1.1.1

Apply the product rule to $\frac{1 - i \sqrt{3}}{2}$ .

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

Step 8.2.7.2.2.1.1.2

Raise $2$ to the power of $4$ .

$x = \frac{{(1 - i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.2

Use the Binomial Theorem.

$x = \frac{1^{4} + 4 \cdot 1^{3} (- i \sqrt{3}) + 6 \cdot 1^{2} {(- i \sqrt{3})}^{2} + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3

Simplify terms.

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Step 8.2.7.2.2.1.3.1

Simplify each term.

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Step 8.2.7.2.2.1.3.1.1

One to any power is one.

$x = \frac{1 + 4 \cdot 1^{3} (- i \sqrt{3}) + 6 \cdot 1^{2} {(- i \sqrt{3})}^{2} + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.2

One to any power is one.

$x = \frac{1 + 4 \cdot 1 (- i \sqrt{3}) + 6 \cdot 1^{2} {(- i \sqrt{3})}^{2} + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.3

Multiply $4$ by $1$ .

$x = \frac{1 + 4 (- i \sqrt{3}) + 6 \cdot 1^{2} {(- i \sqrt{3})}^{2} + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.4

Multiply $- 1$ by $4$ .

$x = \frac{1 - 4 (i \sqrt{3}) + 6 \cdot 1^{2} {(- i \sqrt{3})}^{2} + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.5

One to any power is one.

$x = \frac{1 - 4 i \sqrt{3} + 6 \cdot 1 {(- i \sqrt{3})}^{2} + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.6

Multiply $6$ by $1$ .

$x = \frac{1 - 4 i \sqrt{3} + 6 {(- i \sqrt{3})}^{2} + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.7

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

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Step 8.2.7.2.2.1.3.1.7.1

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

$x = \frac{1 - 4 i \sqrt{3} + 6 ({(- i)}^{2} {\sqrt{3}}^{2}) + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.7.2

Apply the product rule to $- i$ .

$x = \frac{1 - 4 i \sqrt{3} + 6 ({(- 1)}^{2} i^{2} {\sqrt{3}}^{2}) + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.8

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

$x = \frac{1 - 4 i \sqrt{3} + 6 (1 i^{2} {\sqrt{3}}^{2}) + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.9

Multiply $i^{2}$ by $1$ .

$x = \frac{1 - 4 i \sqrt{3} + 6 (i^{2} {\sqrt{3}}^{2}) + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.10

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

$x = \frac{1 - 4 i \sqrt{3} + 6 (- 1 {\sqrt{3}}^{2}) + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.11

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

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Step 8.2.7.2.2.1.3.1.11.1

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

$x = \frac{1 - 4 i \sqrt{3} + 6 (- 1 {(3^{\frac{1}{2}})}^{2}) + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.11.2

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

$x = \frac{1 - 4 i \sqrt{3} + 6 (- 1 \cdot 3^{\frac{1}{2} \cdot 2}) + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.11.3

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

$x = \frac{1 - 4 i \sqrt{3} + 6 (- 1 \cdot 3^{\frac{2}{2}}) + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.11.4

Cancel the common factor of $2$ .

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Step 8.2.7.2.2.1.3.1.11.4.1

Cancel the common factor.

$x = \frac{1 - 4 i \sqrt{3} + 6 (- 1 \cdot 3^{\frac{2}{2}}) + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.11.4.2

Rewrite the expression.

$x = \frac{1 - 4 i \sqrt{3} + 6 (- 1 \cdot 3^{1}) + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.11.5

Evaluate the exponent.

$x = \frac{1 - 4 i \sqrt{3} + 6 (- 1 \cdot 3) + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.12

Multiply $6 (- 1 \cdot 3)$ .

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Step 8.2.7.2.2.1.3.1.12.1

Multiply $- 1$ by $3$ .

$x = \frac{1 - 4 i \sqrt{3} + 6 \cdot - 3 + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.12.2

Multiply $6$ by $- 3$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 \cdot 1 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.13

Multiply $4$ by $1$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 {(- i \sqrt{3})}^{3} + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.14

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

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Step 8.2.7.2.2.1.3.1.14.1

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 ({(- i)}^{3} {\sqrt{3}}^{3}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.14.2

Apply the product rule to $- i$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 ({(- 1)}^{3} i^{3} {\sqrt{3}}^{3}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.15

Raise $- 1$ to the power of $3$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (- i^{3} {\sqrt{3}}^{3}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.16

Factor out $i^{2}$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (- (i^{2} \cdot i) {\sqrt{3}}^{3}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.17

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (- (- 1 \cdot i) {\sqrt{3}}^{3}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.18

Rewrite $- 1 i$ as $- i$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (- - i {\sqrt{3}}^{3}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.19

Multiply $- 1$ by $- 1$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (1 i {\sqrt{3}}^{3}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.20

Multiply $i$ by $1$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (i {\sqrt{3}}^{3}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.21

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (i \sqrt{3^{3}}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.22

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (i \sqrt{27}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.23

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

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Step 8.2.7.2.2.1.3.1.23.1

Factor $9$ out of $27$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (i \sqrt{9 (3)}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.23.2

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (i \sqrt{3^{2} \cdot 3}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.24

Pull terms out from under the radical.

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (i (3 \sqrt{3})) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.25

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 4 (3 \cdot i \sqrt{3}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.26

Multiply $3$ by $4$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 (i \sqrt{3}) + {(- i \sqrt{3})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.27

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

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Step 8.2.7.2.2.1.3.1.27.1

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + {(- i)}^{4} {\sqrt{3}}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.27.2

Apply the product rule to $- i$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + {(- 1)}^{4} i^{4} {\sqrt{3}}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.28

Raise $- 1$ to the power of $4$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + 1 i^{4} {\sqrt{3}}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.29

Multiply $i^{4}$ by $1$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + i^{4} {\sqrt{3}}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.30

Rewrite $i^{4}$ as $1$ .

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Step 8.2.7.2.2.1.3.1.30.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + {(i^{2})}^{2} {\sqrt{3}}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.30.2

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + {(- 1)}^{2} {\sqrt{3}}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.30.3

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + 1 {\sqrt{3}}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.31

Multiply ${\sqrt{3}}^{4}$ by $1$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + {\sqrt{3}}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.32

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 8.2.7.2.2.1.3.1.32.1

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + {(3^{\frac{1}{2}})}^{4}}{16}$

Step 8.2.7.2.2.1.3.1.32.2

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + 3^{\frac{1}{2} \cdot 4}}{16}$

Step 8.2.7.2.2.1.3.1.32.3

Combine $\frac{1}{2}$ and $4$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + 3^{\frac{4}{2}}}{16}$

Step 8.2.7.2.2.1.3.1.32.4

Cancel the common factor of $4$ and $2$ .

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Step 8.2.7.2.2.1.3.1.32.4.1

Factor $2$ out of $4$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + 3^{\frac{2 \cdot 2}{2}}}{16}$

Step 8.2.7.2.2.1.3.1.32.4.2

Cancel the common factors.

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Step 8.2.7.2.2.1.3.1.32.4.2.1

Factor $2$ out of $2$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + 3^{\frac{2 \cdot 2}{2 (1)}}}{16}$

Step 8.2.7.2.2.1.3.1.32.4.2.2

Cancel the common factor.

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + 3^{\frac{2 \cdot 2}{2 \cdot 1}}}{16}$

Step 8.2.7.2.2.1.3.1.32.4.2.3

Rewrite the expression.

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + 3^{\frac{2}{1}}}{16}$

Step 8.2.7.2.2.1.3.1.32.4.2.4

Divide $2$ by $1$ .

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + 3^{2}}{16}$

Step 8.2.7.2.2.1.3.1.33

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

$x = \frac{1 - 4 i \sqrt{3} - 18 + 12 i \sqrt{3} + 9}{16}$

Step 8.2.7.2.2.1.3.2

Simplify terms.

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Step 8.2.7.2.2.1.3.2.1

Subtract $18$ from $1$ .

$x = \frac{- 4 i \sqrt{3} - 17 + 12 i \sqrt{3} + 9}{16}$

Step 8.2.7.2.2.1.3.2.2

Add $- 4 i \sqrt{3}$ and $12 i \sqrt{3}$ .

$x = \frac{8 i \sqrt{3} - 17 + 9}{16}$

Step 8.2.7.2.2.1.3.2.3

Simplify the expression.

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Step 8.2.7.2.2.1.3.2.3.1

Add $- 17$ and $9$ .

$x = \frac{8 i \sqrt{3} - 8}{16}$

Step 8.2.7.2.2.1.3.2.3.2

Reorder $8 i \sqrt{3}$ and $- 8$ .

$x = \frac{- 8 + 8 i \sqrt{3}}{16}$

Step 8.2.7.2.2.1.3.2.4

Cancel the common factor of $- 8 + 8 i \sqrt{3}$ and $16$ .

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Step 8.2.7.2.2.1.3.2.4.1

Factor $8$ out of $- 8$ .

$x = \frac{8 \cdot - 1 + 8 i \sqrt{3}}{16}$

Step 8.2.7.2.2.1.3.2.4.2

Factor $8$ out of $8 i \sqrt{3}$ .

$x = \frac{8 \cdot - 1 + 8 (i \sqrt{3})}{16}$

Step 8.2.7.2.2.1.3.2.4.3

Factor $8$ out of $8 \cdot - 1 + 8 (i \sqrt{3})$ .

$x = \frac{8 \cdot (- 1 + i \sqrt{3})}{16}$

Step 8.2.7.2.2.1.3.2.4.4

Cancel the common factors.

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Step 8.2.7.2.2.1.3.2.4.4.1

Factor $8$ out of $16$ .

$x = \frac{8 \cdot (- 1 + i \sqrt{3})}{8 (2)}$

Step 8.2.7.2.2.1.3.2.4.4.2

Cancel the common factor.

$x = \frac{8 \cdot (- 1 + i \sqrt{3})}{8 \cdot 2}$

Step 8.2.7.2.2.1.3.2.4.4.3

Rewrite the expression.

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

Step 8.2.7.2.2.1.3.2.5

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

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

Step 8.2.7.2.2.1.3.2.6

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

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

Step 8.2.7.2.2.1.3.2.7

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

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

Step 8.2.7.2.2.1.3.2.8

Move the negative in front of the fraction.

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

Step 8.2.8

List all of the solutions.

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

Step 9

Set $x^{\frac{3}{4}} + x^{\frac{3}{2}} + 1$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $x^{\frac{3}{4}} + x^{\frac{3}{2}} + 1$ equal to $0$ .

$x^{\frac{3}{4}} + x^{\frac{3}{2}} + 1 = 0$

Step 9.2

Solve $x^{\frac{3}{4}} + x^{\frac{3}{2}} + 1 = 0$ for $x$ .

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Step 9.2.1

Find a common factor $x^{\frac{3}{4}}$ that is present in each term.

$x^{\frac{3}{4}} {(x^{\frac{3}{4}})}^{2} + 1$

Step 9.2.2

Substitute $u$ for $x^{\frac{3}{4}}$ .

$(u) {(u)}^{2} + 1 = 0$

Step 9.2.3

Solve for $u$ .

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Step 9.2.3.1

Multiply $u$ by ${(u)}^{2}$ by adding the exponents.

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Step 9.2.3.1.1

Multiply $u$ by ${(u)}^{2}$ .

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Step 9.2.3.1.1.1

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

$u^{1} {(u)}^{2} + 1 = 0$

Step 9.2.3.1.1.2

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

$u^{1 + 2} + 1 = 0$

Step 9.2.3.1.2

Add $1$ and $2$ .

$u^{3} + 1 = 0$

Step 9.2.3.2

Subtract $1$ from both sides of the equation.

$u^{3} = - 1$

Step 9.2.3.3

Add $1$ to both sides of the equation.

$u^{3} + 1 = 0$

Step 9.2.3.4

Factor the left side of the equation.

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Step 9.2.3.4.1

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

$u^{3} + 1^{3} = 0$

Step 9.2.3.4.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = u$ and $b = 1$ .

$(u + 1) (u^{2} - u \cdot 1 + 1^{2}) = 0$

Step 9.2.3.4.3

Simplify.

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Step 9.2.3.4.3.1

Multiply $- 1$ by $1$ .

$(u + 1) (u^{2} - u + 1^{2}) = 0$

Step 9.2.3.4.3.2

One to any power is one.

$(u + 1) (u^{2} - u + 1) = 0$

Step 9.2.3.5

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

$u + 1 = 0$

$u^{2} - u + 1 = 0$

Step 9.2.3.6

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 9.2.3.6.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 9.2.3.6.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 9.2.3.7

Set $u^{2} - u + 1$ equal to $0$ and solve for $u$ .

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Step 9.2.3.7.1

Set $u^{2} - u + 1$ equal to $0$ .

$u^{2} - u + 1 = 0$

Step 9.2.3.7.2

Solve $u^{2} - u + 1 = 0$ for $u$ .

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Step 9.2.3.7.2.1

Use the quadratic formula to find the solutions.

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

Step 9.2.3.7.2.2

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

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

Step 9.2.3.7.2.3

Simplify.

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Step 9.2.3.7.2.3.1

Simplify the numerator.

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Step 9.2.3.7.2.3.1.1

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

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

Step 9.2.3.7.2.3.1.2

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

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Step 9.2.3.7.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 9.2.3.7.2.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 9.2.3.7.2.3.1.3

Subtract $4$ from $1$ .

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

Step 9.2.3.7.2.3.1.4

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

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

Step 9.2.3.7.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 9.2.3.7.2.3.1.6

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

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

Step 9.2.3.7.2.3.2

Multiply $2$ by $1$ .

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

Step 9.2.3.7.2.4

The final answer is the combination of both solutions.

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

Step 9.2.3.8

The final solution is all the values that make $(u + 1) (u^{2} - u + 1) = 0$ true.

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

Step 9.2.4

Substitute $x$ for $u$ .

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

Step 9.2.5

Solve for $x^{\frac{3}{4}} = - 1$ for $x$ .

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Step 9.2.5.1

Raise each side of the equation to the power of $\frac{4}{3}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{3}{4}})}^{\frac{4}{3}} = {(- 1)}^{\frac{4}{3}}$

Step 9.2.5.2

Simplify the exponent.

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Step 9.2.5.2.1

Simplify the left side.

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Step 9.2.5.2.1.1

Simplify ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 9.2.5.2.1.1.1

Multiply the exponents in ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 9.2.5.2.1.1.1.1

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

$x^{\frac{3}{4} \cdot \frac{4}{3}} = {(- 1)}^{\frac{4}{3}}$

Step 9.2.5.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 9.2.5.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{3}{4} \cdot \frac{4}{3}} = {(- 1)}^{\frac{4}{3}}$

Step 9.2.5.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{4} \cdot 4} = {(- 1)}^{\frac{4}{3}}$

Step 9.2.5.2.1.1.1.3

Cancel the common factor of $4$ .

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Step 9.2.5.2.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = {(- 1)}^{\frac{4}{3}}$

Step 9.2.5.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = {(- 1)}^{\frac{4}{3}}$

Step 9.2.5.2.1.1.2

Simplify.

$x = {(- 1)}^{\frac{4}{3}}$

Step 9.2.5.2.2

Simplify the right side.

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Step 9.2.5.2.2.1

Simplify ${(- 1)}^{\frac{4}{3}}$ .

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Step 9.2.5.2.2.1.1

Simplify the expression.

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Step 9.2.5.2.2.1.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = {({(- 1)}^{3})}^{\frac{4}{3}}$

Step 9.2.5.2.2.1.1.2

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

$x = {(- 1)}^{3 (\frac{4}{3})}$

Step 9.2.5.2.2.1.2

Cancel the common factor of $3$ .

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Step 9.2.5.2.2.1.2.1

Cancel the common factor.

$x = {(- 1)}^{3 (\frac{4}{3})}$

Step 9.2.5.2.2.1.2.2

Rewrite the expression.

$x = {(- 1)}^{4}$

Step 9.2.5.2.2.1.3

Raise $- 1$ to the power of $4$ .

$x = 1$

Step 9.2.6

Solve for $x^{\frac{3}{4}} = \frac{1 + i \sqrt{3}}{2}$ for $x$ .

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Step 9.2.6.1

Raise each side of the equation to the power of $\frac{4}{3}$ to eliminate the fractional exponent on the left side.

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

Step 9.2.6.2

Simplify the exponent.

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Step 9.2.6.2.1

Simplify the left side.

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Step 9.2.6.2.1.1

Simplify ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 9.2.6.2.1.1.1

Multiply the exponents in ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 9.2.6.2.1.1.1.1

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

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

Step 9.2.6.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 9.2.6.2.1.1.1.2.1

Cancel the common factor.

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

Step 9.2.6.2.1.1.1.2.2

Rewrite the expression.

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

Step 9.2.6.2.1.1.1.3

Cancel the common factor of $4$ .

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Step 9.2.6.2.1.1.1.3.1

Cancel the common factor.

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

Step 9.2.6.2.1.1.1.3.2

Rewrite the expression.

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

Step 9.2.6.2.1.1.2

Simplify.

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

Step 9.2.6.2.2

Simplify the right side.

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Step 9.2.6.2.2.1

Apply the product rule to $\frac{1 + i \sqrt{3}}{2}$ .

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

Step 9.2.7

Solve for $x^{\frac{3}{4}} = \frac{1 - i \sqrt{3}}{2}$ for $x$ .

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Step 9.2.7.1

Raise each side of the equation to the power of $\frac{4}{3}$ to eliminate the fractional exponent on the left side.

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

Step 9.2.7.2

Simplify the exponent.

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Step 9.2.7.2.1

Simplify the left side.

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Step 9.2.7.2.1.1

Simplify ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 9.2.7.2.1.1.1

Multiply the exponents in ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 9.2.7.2.1.1.1.1

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

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

Step 9.2.7.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 9.2.7.2.1.1.1.2.1

Cancel the common factor.

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

Step 9.2.7.2.1.1.1.2.2

Rewrite the expression.

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

Step 9.2.7.2.1.1.1.3

Cancel the common factor of $4$ .

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Step 9.2.7.2.1.1.1.3.1

Cancel the common factor.

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

Step 9.2.7.2.1.1.1.3.2

Rewrite the expression.

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

Step 9.2.7.2.1.1.2

Simplify.

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

Step 9.2.7.2.2

Simplify the right side.

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Step 9.2.7.2.2.1

Apply the product rule to $\frac{1 - i \sqrt{3}}{2}$ .

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

Step 9.2.8

List all of the solutions.

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

Step 10

The final solution is all the values that make $x^{\frac{3}{4}} (1 - x^{\frac{1}{4}}) (x^{\frac{1}{4}} + x^{\frac{1}{2}} + 1) (x^{\frac{3}{4}} + x^{\frac{3}{2}} + 1) = 0$ true.

$x = 0, 1, - \frac{1 + i \sqrt{3}}{2}, - \frac{1 - i \sqrt{3}}{2}, \frac{{(1 + i \sqrt{3})}^{\frac{4}{3}}}{2^{\frac{4}{3}}}, \frac{{(1 - i \sqrt{3})}^{\frac{4}{3}}}{2^{\frac{4}{3}}}$