Algebra Examples

$6 x^{\frac{1}{2}} - 9 x^{\frac{1}{4}} + 2 = 0$

Step 1

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

$6 {(x^{\frac{1}{4}})}^{2} - 9 x^{\frac{1}{4}} + 2$

Step 2

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

$6 {(u)}^{2} - 9 u + 2 = 0$

Step 3

Solve for $u$ .

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Step 3.1

Remove parentheses.

$6 u^{2} - 9 u + 2 = 0$

Step 3.2

Use the quadratic formula to find the solutions.

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

Step 3.3

Substitute the values $a = 6$ , $b = - 9$ , and $c = 2$ into the quadratic formula and solve for $u$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (6 \cdot 2)}}{2 \cdot 6}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

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

$u = \frac{9 \pm \sqrt{81 - 4 \cdot 6 \cdot 2}}{2 \cdot 6}$

Step 3.4.1.2

Multiply $- 4 \cdot 6 \cdot 2$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $6$ .

$u = \frac{9 \pm \sqrt{81 - 24 \cdot 2}}{2 \cdot 6}$

Step 3.4.1.2.2

Multiply $- 24$ by $2$ .

$u = \frac{9 \pm \sqrt{81 - 48}}{2 \cdot 6}$

Step 3.4.1.3

Subtract $48$ from $81$ .

$u = \frac{9 \pm \sqrt{33}}{2 \cdot 6}$

Step 3.4.2

Multiply $2$ by $6$ .

$u = \frac{9 \pm \sqrt{33}}{12}$

Step 3.5

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

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

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

$u = \frac{9 \pm \sqrt{81 - 4 \cdot 6 \cdot 2}}{2 \cdot 6}$

Step 3.5.1.2

Multiply $- 4 \cdot 6 \cdot 2$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $6$ .

$u = \frac{9 \pm \sqrt{81 - 24 \cdot 2}}{2 \cdot 6}$

Step 3.5.1.2.2

Multiply $- 24$ by $2$ .

$u = \frac{9 \pm \sqrt{81 - 48}}{2 \cdot 6}$

Step 3.5.1.3

Subtract $48$ from $81$ .

$u = \frac{9 \pm \sqrt{33}}{2 \cdot 6}$

Step 3.5.2

Multiply $2$ by $6$ .

$u = \frac{9 \pm \sqrt{33}}{12}$

Step 3.5.3

Change the $\pm$ to $+$ .

$u = \frac{9 + \sqrt{33}}{12}$

Step 3.6

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

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

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

$u = \frac{9 \pm \sqrt{81 - 4 \cdot 6 \cdot 2}}{2 \cdot 6}$

Step 3.6.1.2

Multiply $- 4 \cdot 6 \cdot 2$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $6$ .

$u = \frac{9 \pm \sqrt{81 - 24 \cdot 2}}{2 \cdot 6}$

Step 3.6.1.2.2

Multiply $- 24$ by $2$ .

$u = \frac{9 \pm \sqrt{81 - 48}}{2 \cdot 6}$

Step 3.6.1.3

Subtract $48$ from $81$ .

$u = \frac{9 \pm \sqrt{33}}{2 \cdot 6}$

Step 3.6.2

Multiply $2$ by $6$ .

$u = \frac{9 \pm \sqrt{33}}{12}$

Step 3.6.3

Change the $\pm$ to $-$ .

$u = \frac{9 - \sqrt{33}}{12}$

Step 3.7

The final answer is the combination of both solutions.

$u = \frac{9 + \sqrt{33}}{12}, \frac{9 - \sqrt{33}}{12}$

Step 4

Substitute $x$ for $u$ .

$x^{\frac{1}{4}} = \frac{9 + \sqrt{33}}{12}, \frac{9 - \sqrt{33}}{12}$

Step 5

Solve for $x^{\frac{1}{4}} = \frac{9 + \sqrt{33}}{12}$ for $x$ .

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Step 5.1

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

${(x^{\frac{1}{4}})}^{4} = {(\frac{9 + \sqrt{33}}{12})}^{4}$

Step 5.2

Simplify the exponent.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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Step 5.2.1.1.1.1

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

$x^{\frac{1}{4} \cdot 4} = {(\frac{9 + \sqrt{33}}{12})}^{4}$

Step 5.2.1.1.1.2

Cancel the common factor of $4$ .

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Step 5.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = {(\frac{9 + \sqrt{33}}{12})}^{4}$

Step 5.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(\frac{9 + \sqrt{33}}{12})}^{4}$

Step 5.2.1.1.2

Simplify.

$x = {(\frac{9 + \sqrt{33}}{12})}^{4}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify ${(\frac{9 + \sqrt{33}}{12})}^{4}$ .

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Step 5.2.2.1.1

Simplify the expression.

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Step 5.2.2.1.1.1

Apply the product rule to $\frac{9 + \sqrt{33}}{12}$ .

$x = \frac{{(9 + \sqrt{33})}^{4}}{12^{4}}$

Step 5.2.2.1.1.2

Raise $12$ to the power of $4$ .

$x = \frac{{(9 + \sqrt{33})}^{4}}{20736}$

Step 5.2.2.1.2

Use the Binomial Theorem.

$x = \frac{9^{4} + 4 \cdot 9^{3} \sqrt{33} + 6 \cdot 9^{2} {\sqrt{33}}^{2} + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3

Simplify terms.

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Step 5.2.2.1.3.1

Simplify each term.

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Step 5.2.2.1.3.1.1

Raise $9$ to the power of $4$ .

$x = \frac{6561 + 4 \cdot 9^{3} \sqrt{33} + 6 \cdot 9^{2} {\sqrt{33}}^{2} + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.2

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

$x = \frac{6561 + 4 \cdot 729 \sqrt{33} + 6 \cdot 9^{2} {\sqrt{33}}^{2} + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.3

Multiply $4$ by $729$ .

$x = \frac{6561 + 2916 \sqrt{33} + 6 \cdot 9^{2} {\sqrt{33}}^{2} + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.4

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

$x = \frac{6561 + 2916 \sqrt{33} + 6 \cdot 81 {\sqrt{33}}^{2} + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.5

Multiply $6$ by $81$ .

$x = \frac{6561 + 2916 \sqrt{33} + 486 {\sqrt{33}}^{2} + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.6

Rewrite ${\sqrt{33}}^{2}$ as $33$ .

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Step 5.2.2.1.3.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$x = \frac{6561 + 2916 \sqrt{33} + 486 {(33^{\frac{1}{2}})}^{2} + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.6.2

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

$x = \frac{6561 + 2916 \sqrt{33} + 486 \cdot 33^{\frac{1}{2} \cdot 2} + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.6.3

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

$x = \frac{6561 + 2916 \sqrt{33} + 486 \cdot 33^{\frac{2}{2}} + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.6.4

Cancel the common factor of $2$ .

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Step 5.2.2.1.3.1.6.4.1

Cancel the common factor.

$x = \frac{6561 + 2916 \sqrt{33} + 486 \cdot 33^{\frac{2}{2}} + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.6.4.2

Rewrite the expression.

$x = \frac{6561 + 2916 \sqrt{33} + 486 \cdot 33^{1} + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.6.5

Evaluate the exponent.

$x = \frac{6561 + 2916 \sqrt{33} + 486 \cdot 33 + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.7

Multiply $486$ by $33$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 4 \cdot 9 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.8

Multiply $4$ by $9$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 36 {\sqrt{33}}^{3} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.9

Rewrite ${\sqrt{33}}^{3}$ as $\sqrt{33^{3}}$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 36 \sqrt{33^{3}} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.10

Raise $33$ to the power of $3$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 36 \sqrt{35937} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.11

Rewrite $35937$ as $33^{2} \cdot 33$ .

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Step 5.2.2.1.3.1.11.1

Factor $1089$ out of $35937$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 36 \sqrt{1089 (33)} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.11.2

Rewrite $1089$ as $33^{2}$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 36 \sqrt{33^{2} \cdot 33} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.12

Pull terms out from under the radical.

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 36 (33 \sqrt{33}) + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.13

Multiply $33$ by $36$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 1188 \sqrt{33} + {\sqrt{33}}^{4}}{20736}$

Step 5.2.2.1.3.1.14

Rewrite ${\sqrt{33}}^{4}$ as $33^{2}$ .

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Step 5.2.2.1.3.1.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 1188 \sqrt{33} + {(33^{\frac{1}{2}})}^{4}}{20736}$

Step 5.2.2.1.3.1.14.2

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

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 1188 \sqrt{33} + 33^{\frac{1}{2} \cdot 4}}{20736}$

Step 5.2.2.1.3.1.14.3

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

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 1188 \sqrt{33} + 33^{\frac{4}{2}}}{20736}$

Step 5.2.2.1.3.1.14.4

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

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Step 5.2.2.1.3.1.14.4.1

Factor $2$ out of $4$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 1188 \sqrt{33} + 33^{\frac{2 \cdot 2}{2}}}{20736}$

Step 5.2.2.1.3.1.14.4.2

Cancel the common factors.

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Step 5.2.2.1.3.1.14.4.2.1

Factor $2$ out of $2$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 1188 \sqrt{33} + 33^{\frac{2 \cdot 2}{2 (1)}}}{20736}$

Step 5.2.2.1.3.1.14.4.2.2

Cancel the common factor.

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 1188 \sqrt{33} + 33^{\frac{2 \cdot 2}{2 \cdot 1}}}{20736}$

Step 5.2.2.1.3.1.14.4.2.3

Rewrite the expression.

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 1188 \sqrt{33} + 33^{\frac{2}{1}}}{20736}$

Step 5.2.2.1.3.1.14.4.2.4

Divide $2$ by $1$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 1188 \sqrt{33} + 33^{2}}{20736}$

Step 5.2.2.1.3.1.15

Raise $33$ to the power of $2$ .

$x = \frac{6561 + 2916 \sqrt{33} + 16038 + 1188 \sqrt{33} + 1089}{20736}$

Step 5.2.2.1.3.2

Simplify terms.

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Step 5.2.2.1.3.2.1

Add $6561$ and $16038$ .

$x = \frac{22599 + 2916 \sqrt{33} + 1188 \sqrt{33} + 1089}{20736}$

Step 5.2.2.1.3.2.2

Add $22599$ and $1089$ .

$x = \frac{23688 + 2916 \sqrt{33} + 1188 \sqrt{33}}{20736}$

Step 5.2.2.1.3.2.3

Add $2916 \sqrt{33}$ and $1188 \sqrt{33}$ .

$x = \frac{23688 + 4104 \sqrt{33}}{20736}$

Step 5.2.2.1.3.2.4

Cancel the common factor of $23688 + 4104 \sqrt{33}$ and $20736$ .

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Step 5.2.2.1.3.2.4.1

Factor $72$ out of $23688$ .

$x = \frac{72 (329) + 4104 \sqrt{33}}{20736}$

Step 5.2.2.1.3.2.4.2

Factor $72$ out of $4104 \sqrt{33}$ .

$x = \frac{72 (329) + 72 (57 \sqrt{33})}{20736}$

Step 5.2.2.1.3.2.4.3

Factor $72$ out of $72 (329) + 72 (57 \sqrt{33})$ .

$x = \frac{72 (329 + 57 \sqrt{33})}{20736}$

Step 5.2.2.1.3.2.4.4

Cancel the common factors.

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Step 5.2.2.1.3.2.4.4.1

Factor $72$ out of $20736$ .

$x = \frac{72 (329 + 57 \sqrt{33})}{72 \cdot 288}$

Step 5.2.2.1.3.2.4.4.2

Cancel the common factor.

$x = \frac{72 (329 + 57 \sqrt{33})}{72 \cdot 288}$

Step 5.2.2.1.3.2.4.4.3

Rewrite the expression.

$x = \frac{329 + 57 \sqrt{33}}{288}$

Step 6

Solve for $x^{\frac{1}{4}} = \frac{9 - \sqrt{33}}{12}$ for $x$ .

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Step 6.1

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

${(x^{\frac{1}{4}})}^{4} = {(\frac{9 - \sqrt{33}}{12})}^{4}$

Step 6.2

Simplify the exponent.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

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

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Step 6.2.1.1.1

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

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Step 6.2.1.1.1.1

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

$x^{\frac{1}{4} \cdot 4} = {(\frac{9 - \sqrt{33}}{12})}^{4}$

Step 6.2.1.1.1.2

Cancel the common factor of $4$ .

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Step 6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = {(\frac{9 - \sqrt{33}}{12})}^{4}$

Step 6.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(\frac{9 - \sqrt{33}}{12})}^{4}$

Step 6.2.1.1.2

Simplify.

$x = {(\frac{9 - \sqrt{33}}{12})}^{4}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify ${(\frac{9 - \sqrt{33}}{12})}^{4}$ .

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Step 6.2.2.1.1

Simplify the expression.

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Step 6.2.2.1.1.1

Apply the product rule to $\frac{9 - \sqrt{33}}{12}$ .

$x = \frac{{(9 - \sqrt{33})}^{4}}{12^{4}}$

Step 6.2.2.1.1.2

Raise $12$ to the power of $4$ .

$x = \frac{{(9 - \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.2

Use the Binomial Theorem.

$x = \frac{9^{4} + 4 \cdot 9^{3} (- \sqrt{33}) + 6 \cdot 9^{2} {(- \sqrt{33})}^{2} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3

Simplify terms.

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Step 6.2.2.1.3.1

Simplify each term.

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Step 6.2.2.1.3.1.1

Raise $9$ to the power of $4$ .

$x = \frac{6561 + 4 \cdot 9^{3} (- \sqrt{33}) + 6 \cdot 9^{2} {(- \sqrt{33})}^{2} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.2

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

$x = \frac{6561 + 4 \cdot 729 (- \sqrt{33}) + 6 \cdot 9^{2} {(- \sqrt{33})}^{2} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.3

Multiply $4$ by $729$ .

$x = \frac{6561 + 2916 (- \sqrt{33}) + 6 \cdot 9^{2} {(- \sqrt{33})}^{2} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.4

Multiply $- 1$ by $2916$ .

$x = \frac{6561 - 2916 \sqrt{33} + 6 \cdot 9^{2} {(- \sqrt{33})}^{2} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.5

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

$x = \frac{6561 - 2916 \sqrt{33} + 6 \cdot 81 {(- \sqrt{33})}^{2} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.6

Multiply $6$ by $81$ .

$x = \frac{6561 - 2916 \sqrt{33} + 486 {(- \sqrt{33})}^{2} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.7

Apply the product rule to $- \sqrt{33}$ .

$x = \frac{6561 - 2916 \sqrt{33} + 486 ({(- 1)}^{2} {\sqrt{33}}^{2}) + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.8

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

$x = \frac{6561 - 2916 \sqrt{33} + 486 (1 {\sqrt{33}}^{2}) + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.9

Multiply ${\sqrt{33}}^{2}$ by $1$ .

$x = \frac{6561 - 2916 \sqrt{33} + 486 {\sqrt{33}}^{2} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.10

Rewrite ${\sqrt{33}}^{2}$ as $33$ .

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Step 6.2.2.1.3.1.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$x = \frac{6561 - 2916 \sqrt{33} + 486 {(33^{\frac{1}{2}})}^{2} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.10.2

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

$x = \frac{6561 - 2916 \sqrt{33} + 486 \cdot 33^{\frac{1}{2} \cdot 2} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.10.3

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

$x = \frac{6561 - 2916 \sqrt{33} + 486 \cdot 33^{\frac{2}{2}} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.10.4

Cancel the common factor of $2$ .

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Step 6.2.2.1.3.1.10.4.1

Cancel the common factor.

$x = \frac{6561 - 2916 \sqrt{33} + 486 \cdot 33^{\frac{2}{2}} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.10.4.2

Rewrite the expression.

$x = \frac{6561 - 2916 \sqrt{33} + 486 \cdot 33^{1} + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.10.5

Evaluate the exponent.

$x = \frac{6561 - 2916 \sqrt{33} + 486 \cdot 33 + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.11

Multiply $486$ by $33$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 + 4 \cdot 9 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.12

Multiply $4$ by $9$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 + 36 {(- \sqrt{33})}^{3} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.13

Apply the product rule to $- \sqrt{33}$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 + 36 ({(- 1)}^{3} {\sqrt{33}}^{3}) + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.14

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

$x = \frac{6561 - 2916 \sqrt{33} + 16038 + 36 (- {\sqrt{33}}^{3}) + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.15

Rewrite ${\sqrt{33}}^{3}$ as $\sqrt{33^{3}}$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 + 36 (- \sqrt{33^{3}}) + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.16

Raise $33$ to the power of $3$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 + 36 (- \sqrt{35937}) + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.17

Rewrite $35937$ as $33^{2} \cdot 33$ .

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Step 6.2.2.1.3.1.17.1

Factor $1089$ out of $35937$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 + 36 (- \sqrt{1089 (33)}) + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.17.2

Rewrite $1089$ as $33^{2}$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 + 36 (- \sqrt{33^{2} \cdot 33}) + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.18

Pull terms out from under the radical.

$x = \frac{6561 - 2916 \sqrt{33} + 16038 + 36 (- (33 \sqrt{33})) + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.19

Multiply $33$ by $- 1$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 + 36 (- 33 \sqrt{33}) + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.20

Multiply $- 33$ by $36$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + {(- \sqrt{33})}^{4}}{20736}$

Step 6.2.2.1.3.1.21

Apply the product rule to $- \sqrt{33}$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + {(- 1)}^{4} {\sqrt{33}}^{4}}{20736}$

Step 6.2.2.1.3.1.22

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

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + 1 {\sqrt{33}}^{4}}{20736}$

Step 6.2.2.1.3.1.23

Multiply ${\sqrt{33}}^{4}$ by $1$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + {\sqrt{33}}^{4}}{20736}$

Step 6.2.2.1.3.1.24

Rewrite ${\sqrt{33}}^{4}$ as $33^{2}$ .

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Step 6.2.2.1.3.1.24.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + {(33^{\frac{1}{2}})}^{4}}{20736}$

Step 6.2.2.1.3.1.24.2

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

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + 33^{\frac{1}{2} \cdot 4}}{20736}$

Step 6.2.2.1.3.1.24.3

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

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + 33^{\frac{4}{2}}}{20736}$

Step 6.2.2.1.3.1.24.4

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

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Step 6.2.2.1.3.1.24.4.1

Factor $2$ out of $4$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + 33^{\frac{2 \cdot 2}{2}}}{20736}$

Step 6.2.2.1.3.1.24.4.2

Cancel the common factors.

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Step 6.2.2.1.3.1.24.4.2.1

Factor $2$ out of $2$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + 33^{\frac{2 \cdot 2}{2 (1)}}}{20736}$

Step 6.2.2.1.3.1.24.4.2.2

Cancel the common factor.

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + 33^{\frac{2 \cdot 2}{2 \cdot 1}}}{20736}$

Step 6.2.2.1.3.1.24.4.2.3

Rewrite the expression.

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + 33^{\frac{2}{1}}}{20736}$

Step 6.2.2.1.3.1.24.4.2.4

Divide $2$ by $1$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + 33^{2}}{20736}$

Step 6.2.2.1.3.1.25

Raise $33$ to the power of $2$ .

$x = \frac{6561 - 2916 \sqrt{33} + 16038 - 1188 \sqrt{33} + 1089}{20736}$

Step 6.2.2.1.3.2

Simplify terms.

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Step 6.2.2.1.3.2.1

Add $6561$ and $16038$ .

$x = \frac{22599 - 2916 \sqrt{33} - 1188 \sqrt{33} + 1089}{20736}$

Step 6.2.2.1.3.2.2

Add $22599$ and $1089$ .

$x = \frac{23688 - 2916 \sqrt{33} - 1188 \sqrt{33}}{20736}$

Step 6.2.2.1.3.2.3

Subtract $1188 \sqrt{33}$ from $- 2916 \sqrt{33}$ .

$x = \frac{23688 - 4104 \sqrt{33}}{20736}$

Step 6.2.2.1.3.2.4

Cancel the common factor of $23688 - 4104 \sqrt{33}$ and $20736$ .

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Step 6.2.2.1.3.2.4.1

Factor $72$ out of $23688$ .

$x = \frac{72 (329) - 4104 \sqrt{33}}{20736}$

Step 6.2.2.1.3.2.4.2

Factor $72$ out of $- 4104 \sqrt{33}$ .

$x = \frac{72 (329) + 72 (- 57 \sqrt{33})}{20736}$

Step 6.2.2.1.3.2.4.3

Factor $72$ out of $72 (329) + 72 (- 57 \sqrt{33})$ .

$x = \frac{72 (329 - 57 \sqrt{33})}{20736}$

Step 6.2.2.1.3.2.4.4

Cancel the common factors.

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Step 6.2.2.1.3.2.4.4.1

Factor $72$ out of $20736$ .

$x = \frac{72 (329 - 57 \sqrt{33})}{72 \cdot 288}$

Step 6.2.2.1.3.2.4.4.2

Cancel the common factor.

$x = \frac{72 (329 - 57 \sqrt{33})}{72 \cdot 288}$

Step 6.2.2.1.3.2.4.4.3

Rewrite the expression.

$x = \frac{329 - 57 \sqrt{33}}{288}$

Step 7

List all of the solutions.

$x = \frac{329 + 57 \sqrt{33}}{288}, \frac{329 - 57 \sqrt{33}}{288}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{329 + 57 \sqrt{33}}{288}, \frac{329 - 57 \sqrt{33}}{288}$

Decimal Form:

$x = 2.27930580 \dots, 0.00541642 \dots$