Precalculus Examples

$6 a^{\frac{2}{3}} - a^{\frac{1}{3}} - 20 = 0$

Step 1

Find a common factor $a^{\frac{1}{3}}$ that is present in each term.

$6 {(a^{\frac{1}{3}})}^{2} - a^{\frac{1}{3}} - 20$

Step 2

Substitute $u$ for $a^{\frac{1}{3}}$ .

$6 {(u)}^{2} - (u) - 20 = 0$

Step 3

Solve for $u$ .

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Step 3.1

Multiply $- 1$ by $u$ .

$6 u^{2} - u - 20 = 0$

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 $a = 6$ , $b = - 1$ , and $c = - 20$ into the quadratic formula and solve for $u$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (6 \cdot - 20)}}{2 \cdot 6}$

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

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 6 \cdot - 20}}{2 \cdot 6}$

Step 3.4.1.2

Multiply $- 4 \cdot 6 \cdot - 20$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $6$ .

$u = \frac{1 \pm \sqrt{1 - 24 \cdot - 20}}{2 \cdot 6}$

Step 3.4.1.2.2

Multiply $- 24$ by $- 20$ .

$u = \frac{1 \pm \sqrt{1 + 480}}{2 \cdot 6}$

Step 3.4.1.3

Add $1$ and $480$ .

$u = \frac{1 \pm \sqrt{481}}{2 \cdot 6}$

Step 3.4.2

Multiply $2$ by $6$ .

$u = \frac{1 \pm \sqrt{481}}{12}$

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

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 6 \cdot - 20}}{2 \cdot 6}$

Step 3.5.1.2

Multiply $- 4 \cdot 6 \cdot - 20$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $6$ .

$u = \frac{1 \pm \sqrt{1 - 24 \cdot - 20}}{2 \cdot 6}$

Step 3.5.1.2.2

Multiply $- 24$ by $- 20$ .

$u = \frac{1 \pm \sqrt{1 + 480}}{2 \cdot 6}$

Step 3.5.1.3

Add $1$ and $480$ .

$u = \frac{1 \pm \sqrt{481}}{2 \cdot 6}$

Step 3.5.2

Multiply $2$ by $6$ .

$u = \frac{1 \pm \sqrt{481}}{12}$

Step 3.5.3

Change the $\pm$ to $+$ .

$u = \frac{1 + \sqrt{481}}{12}$

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

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 6 \cdot - 20}}{2 \cdot 6}$

Step 3.6.1.2

Multiply $- 4 \cdot 6 \cdot - 20$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $6$ .

$u = \frac{1 \pm \sqrt{1 - 24 \cdot - 20}}{2 \cdot 6}$

Step 3.6.1.2.2

Multiply $- 24$ by $- 20$ .

$u = \frac{1 \pm \sqrt{1 + 480}}{2 \cdot 6}$

Step 3.6.1.3

Add $1$ and $480$ .

$u = \frac{1 \pm \sqrt{481}}{2 \cdot 6}$

Step 3.6.2

Multiply $2$ by $6$ .

$u = \frac{1 \pm \sqrt{481}}{12}$

Step 3.6.3

Change the $\pm$ to $-$ .

$u = \frac{1 - \sqrt{481}}{12}$

Step 3.7

The final answer is the combination of both solutions.

$u = \frac{1 + \sqrt{481}}{12}, \frac{1 - \sqrt{481}}{12}$

Step 4

Substitute $a$ for $u$ .

$a^{\frac{1}{3}} = \frac{1 + \sqrt{481}}{12}, \frac{1 - \sqrt{481}}{12}$

Step 5

Solve for $a^{\frac{1}{3}} = \frac{1 + \sqrt{481}}{12}$ for $a$ .

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Step 5.1

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(a^{\frac{1}{3}})}^{3} = {(\frac{1 + \sqrt{481}}{12})}^{3}$

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 ${(a^{\frac{1}{3}})}^{3}$ .

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Step 5.2.1.1.1

Multiply the exponents in ${(a^{\frac{1}{3}})}^{3}$ .

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Step 5.2.1.1.1.1

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

$a^{\frac{1}{3} \cdot 3} = {(\frac{1 + \sqrt{481}}{12})}^{3}$

Step 5.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.1.1.2.1

Cancel the common factor.

$a^{\frac{1}{3} \cdot 3} = {(\frac{1 + \sqrt{481}}{12})}^{3}$

Step 5.2.1.1.1.2.2

Rewrite the expression.

$a^{1} = {(\frac{1 + \sqrt{481}}{12})}^{3}$

Step 5.2.1.1.2

Simplify.

$a = {(\frac{1 + \sqrt{481}}{12})}^{3}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify ${(\frac{1 + \sqrt{481}}{12})}^{3}$ .

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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 $\frac{1 + \sqrt{481}}{12}$ .

$a = \frac{{(1 + \sqrt{481})}^{3}}{12^{3}}$

Step 5.2.2.1.1.2

Raise $12$ to the power of $3$ .

$a = \frac{{(1 + \sqrt{481})}^{3}}{1728}$

Step 5.2.2.1.2

Use the Binomial Theorem.

$a = \frac{1^{3} + 3 \cdot 1^{2} \sqrt{481} + 3 \cdot 1 {\sqrt{481}}^{2} + {\sqrt{481}}^{3}}{1728}$

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

One to any power is one.

$a = \frac{1 + 3 \cdot 1^{2} \sqrt{481} + 3 \cdot 1 {\sqrt{481}}^{2} + {\sqrt{481}}^{3}}{1728}$

Step 5.2.2.1.3.1.2

One to any power is one.

$a = \frac{1 + 3 \cdot 1 \sqrt{481} + 3 \cdot 1 {\sqrt{481}}^{2} + {\sqrt{481}}^{3}}{1728}$

Step 5.2.2.1.3.1.3

Multiply $3$ by $1$ .

$a = \frac{1 + 3 \sqrt{481} + 3 \cdot 1 {\sqrt{481}}^{2} + {\sqrt{481}}^{3}}{1728}$

Step 5.2.2.1.3.1.4

Multiply $3$ by $1$ .

$a = \frac{1 + 3 \sqrt{481} + 3 {\sqrt{481}}^{2} + {\sqrt{481}}^{3}}{1728}$

Step 5.2.2.1.3.1.5

Rewrite ${\sqrt{481}}^{2}$ as $481$ .

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Step 5.2.2.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{481}$ as $481^{\frac{1}{2}}$ .

$a = \frac{1 + 3 \sqrt{481} + 3 {(481^{\frac{1}{2}})}^{2} + {\sqrt{481}}^{3}}{1728}$

Step 5.2.2.1.3.1.5.2

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

$a = \frac{1 + 3 \sqrt{481} + 3 \cdot 481^{\frac{1}{2} \cdot 2} + {\sqrt{481}}^{3}}{1728}$

Step 5.2.2.1.3.1.5.3

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

$a = \frac{1 + 3 \sqrt{481} + 3 \cdot 481^{\frac{2}{2}} + {\sqrt{481}}^{3}}{1728}$

Step 5.2.2.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 5.2.2.1.3.1.5.4.1

Cancel the common factor.

$a = \frac{1 + 3 \sqrt{481} + 3 \cdot 481^{\frac{2}{2}} + {\sqrt{481}}^{3}}{1728}$

Step 5.2.2.1.3.1.5.4.2

Rewrite the expression.

$a = \frac{1 + 3 \sqrt{481} + 3 \cdot 481^{1} + {\sqrt{481}}^{3}}{1728}$

Step 5.2.2.1.3.1.5.5

Evaluate the exponent.

$a = \frac{1 + 3 \sqrt{481} + 3 \cdot 481 + {\sqrt{481}}^{3}}{1728}$

Step 5.2.2.1.3.1.6

Multiply $3$ by $481$ .

$a = \frac{1 + 3 \sqrt{481} + 1443 + {\sqrt{481}}^{3}}{1728}$

Step 5.2.2.1.3.1.7

Rewrite ${\sqrt{481}}^{3}$ as $\sqrt{481^{3}}$ .

$a = \frac{1 + 3 \sqrt{481} + 1443 + \sqrt{481^{3}}}{1728}$

Step 5.2.2.1.3.1.8

Raise $481$ to the power of $3$ .

$a = \frac{1 + 3 \sqrt{481} + 1443 + \sqrt{111284641}}{1728}$

Step 5.2.2.1.3.1.9

Rewrite $111284641$ as $481^{2} \cdot 481$ .

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Step 5.2.2.1.3.1.9.1

Factor $231361$ out of $111284641$ .

$a = \frac{1 + 3 \sqrt{481} + 1443 + \sqrt{231361 (481)}}{1728}$

Step 5.2.2.1.3.1.9.2

Rewrite $231361$ as $481^{2}$ .

$a = \frac{1 + 3 \sqrt{481} + 1443 + \sqrt{481^{2} \cdot 481}}{1728}$

Step 5.2.2.1.3.1.10

Pull terms out from under the radical.

$a = \frac{1 + 3 \sqrt{481} + 1443 + 481 \sqrt{481}}{1728}$

Step 5.2.2.1.3.2

Simplify terms.

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Step 5.2.2.1.3.2.1

Add $1$ and $1443$ .

$a = \frac{1444 + 3 \sqrt{481} + 481 \sqrt{481}}{1728}$

Step 5.2.2.1.3.2.2

Add $3 \sqrt{481}$ and $481 \sqrt{481}$ .

$a = \frac{1444 + 484 \sqrt{481}}{1728}$

Step 5.2.2.1.3.2.3

Cancel the common factor of $1444 + 484 \sqrt{481}$ and $1728$ .

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Step 5.2.2.1.3.2.3.1

Factor $4$ out of $1444$ .

$a = \frac{4 (361) + 484 \sqrt{481}}{1728}$

Step 5.2.2.1.3.2.3.2

Factor $4$ out of $484 \sqrt{481}$ .

$a = \frac{4 (361) + 4 (121 \sqrt{481})}{1728}$

Step 5.2.2.1.3.2.3.3

Factor $4$ out of $4 (361) + 4 (121 \sqrt{481})$ .

$a = \frac{4 (361 + 121 \sqrt{481})}{1728}$

Step 5.2.2.1.3.2.3.4

Cancel the common factors.

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Step 5.2.2.1.3.2.3.4.1

Factor $4$ out of $1728$ .

$a = \frac{4 (361 + 121 \sqrt{481})}{4 \cdot 432}$

Step 5.2.2.1.3.2.3.4.2

Cancel the common factor.

$a = \frac{4 (361 + 121 \sqrt{481})}{4 \cdot 432}$

Step 5.2.2.1.3.2.3.4.3

Rewrite the expression.

$a = \frac{361 + 121 \sqrt{481}}{432}$

Step 6

Solve for $a^{\frac{1}{3}} = \frac{1 - \sqrt{481}}{12}$ for $a$ .

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Step 6.1

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(a^{\frac{1}{3}})}^{3} = {(\frac{1 - \sqrt{481}}{12})}^{3}$

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 ${(a^{\frac{1}{3}})}^{3}$ .

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Step 6.2.1.1.1

Multiply the exponents in ${(a^{\frac{1}{3}})}^{3}$ .

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Step 6.2.1.1.1.1

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

$a^{\frac{1}{3} \cdot 3} = {(\frac{1 - \sqrt{481}}{12})}^{3}$

Step 6.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 6.2.1.1.1.2.1

Cancel the common factor.

$a^{\frac{1}{3} \cdot 3} = {(\frac{1 - \sqrt{481}}{12})}^{3}$

Step 6.2.1.1.1.2.2

Rewrite the expression.

$a^{1} = {(\frac{1 - \sqrt{481}}{12})}^{3}$

Step 6.2.1.1.2

Simplify.

$a = {(\frac{1 - \sqrt{481}}{12})}^{3}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify ${(\frac{1 - \sqrt{481}}{12})}^{3}$ .

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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 $\frac{1 - \sqrt{481}}{12}$ .

$a = \frac{{(1 - \sqrt{481})}^{3}}{12^{3}}$

Step 6.2.2.1.1.2

Raise $12$ to the power of $3$ .

$a = \frac{{(1 - \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.2

Use the Binomial Theorem.

$a = \frac{1^{3} + 3 \cdot 1^{2} (- \sqrt{481}) + 3 \cdot 1 {(- \sqrt{481})}^{2} + {(- \sqrt{481})}^{3}}{1728}$

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

One to any power is one.

$a = \frac{1 + 3 \cdot 1^{2} (- \sqrt{481}) + 3 \cdot 1 {(- \sqrt{481})}^{2} + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.2

One to any power is one.

$a = \frac{1 + 3 \cdot 1 (- \sqrt{481}) + 3 \cdot 1 {(- \sqrt{481})}^{2} + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.3

Multiply $3$ by $1$ .

$a = \frac{1 + 3 (- \sqrt{481}) + 3 \cdot 1 {(- \sqrt{481})}^{2} + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.4

Multiply $- 1$ by $3$ .

$a = \frac{1 - 3 \sqrt{481} + 3 \cdot 1 {(- \sqrt{481})}^{2} + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.5

Multiply $3$ by $1$ .

$a = \frac{1 - 3 \sqrt{481} + 3 {(- \sqrt{481})}^{2} + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.6

Apply the product rule to $- \sqrt{481}$ .

$a = \frac{1 - 3 \sqrt{481} + 3 ({(- 1)}^{2} {\sqrt{481}}^{2}) + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.7

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

$a = \frac{1 - 3 \sqrt{481} + 3 (1 {\sqrt{481}}^{2}) + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.8

Multiply ${\sqrt{481}}^{2}$ by $1$ .

$a = \frac{1 - 3 \sqrt{481} + 3 {\sqrt{481}}^{2} + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.9

Rewrite ${\sqrt{481}}^{2}$ as $481$ .

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Step 6.2.2.1.3.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{481}$ as $481^{\frac{1}{2}}$ .

$a = \frac{1 - 3 \sqrt{481} + 3 {(481^{\frac{1}{2}})}^{2} + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.9.2

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

$a = \frac{1 - 3 \sqrt{481} + 3 \cdot 481^{\frac{1}{2} \cdot 2} + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.9.3

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

$a = \frac{1 - 3 \sqrt{481} + 3 \cdot 481^{\frac{2}{2}} + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.9.4

Cancel the common factor of $2$ .

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Step 6.2.2.1.3.1.9.4.1

Cancel the common factor.

$a = \frac{1 - 3 \sqrt{481} + 3 \cdot 481^{\frac{2}{2}} + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.9.4.2

Rewrite the expression.

$a = \frac{1 - 3 \sqrt{481} + 3 \cdot 481^{1} + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.9.5

Evaluate the exponent.

$a = \frac{1 - 3 \sqrt{481} + 3 \cdot 481 + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.10

Multiply $3$ by $481$ .

$a = \frac{1 - 3 \sqrt{481} + 1443 + {(- \sqrt{481})}^{3}}{1728}$

Step 6.2.2.1.3.1.11

Apply the product rule to $- \sqrt{481}$ .

$a = \frac{1 - 3 \sqrt{481} + 1443 + {(- 1)}^{3} {\sqrt{481}}^{3}}{1728}$

Step 6.2.2.1.3.1.12

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

$a = \frac{1 - 3 \sqrt{481} + 1443 - {\sqrt{481}}^{3}}{1728}$

Step 6.2.2.1.3.1.13

Rewrite ${\sqrt{481}}^{3}$ as $\sqrt{481^{3}}$ .

$a = \frac{1 - 3 \sqrt{481} + 1443 - \sqrt{481^{3}}}{1728}$

Step 6.2.2.1.3.1.14

Raise $481$ to the power of $3$ .

$a = \frac{1 - 3 \sqrt{481} + 1443 - \sqrt{111284641}}{1728}$

Step 6.2.2.1.3.1.15

Rewrite $111284641$ as $481^{2} \cdot 481$ .

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Step 6.2.2.1.3.1.15.1

Factor $231361$ out of $111284641$ .

$a = \frac{1 - 3 \sqrt{481} + 1443 - \sqrt{231361 (481)}}{1728}$

Step 6.2.2.1.3.1.15.2

Rewrite $231361$ as $481^{2}$ .

$a = \frac{1 - 3 \sqrt{481} + 1443 - \sqrt{481^{2} \cdot 481}}{1728}$

Step 6.2.2.1.3.1.16

Pull terms out from under the radical.

$a = \frac{1 - 3 \sqrt{481} + 1443 - (481 \sqrt{481})}{1728}$

Step 6.2.2.1.3.1.17

Multiply $481$ by $- 1$ .

$a = \frac{1 - 3 \sqrt{481} + 1443 - 481 \sqrt{481}}{1728}$

Step 6.2.2.1.3.2

Simplify terms.

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Step 6.2.2.1.3.2.1

Add $1$ and $1443$ .

$a = \frac{1444 - 3 \sqrt{481} - 481 \sqrt{481}}{1728}$

Step 6.2.2.1.3.2.2

Subtract $481 \sqrt{481}$ from $- 3 \sqrt{481}$ .

$a = \frac{1444 - 484 \sqrt{481}}{1728}$

Step 6.2.2.1.3.2.3

Cancel the common factor of $1444 - 484 \sqrt{481}$ and $1728$ .

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Step 6.2.2.1.3.2.3.1

Factor $4$ out of $1444$ .

$a = \frac{4 (361) - 484 \sqrt{481}}{1728}$

Step 6.2.2.1.3.2.3.2

Factor $4$ out of $- 484 \sqrt{481}$ .

$a = \frac{4 (361) + 4 (- 121 \sqrt{481})}{1728}$

Step 6.2.2.1.3.2.3.3

Factor $4$ out of $4 (361) + 4 (- 121 \sqrt{481})$ .

$a = \frac{4 (361 - 121 \sqrt{481})}{1728}$

Step 6.2.2.1.3.2.3.4

Cancel the common factors.

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Step 6.2.2.1.3.2.3.4.1

Factor $4$ out of $1728$ .

$a = \frac{4 (361 - 121 \sqrt{481})}{4 \cdot 432}$

Step 6.2.2.1.3.2.3.4.2

Cancel the common factor.

$a = \frac{4 (361 - 121 \sqrt{481})}{4 \cdot 432}$

Step 6.2.2.1.3.2.3.4.3

Rewrite the expression.

$a = \frac{361 - 121 \sqrt{481}}{432}$

Step 7

List all of the solutions.

$a = \frac{361 + 121 \sqrt{481}}{432}, \frac{361 - 121 \sqrt{481}}{432}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$a = \frac{361 + 121 \sqrt{481}}{432}, \frac{361 - 121 \sqrt{481}}{432}$

Decimal Form:

$a = 6.97855827 \dots, - 5.30726198 \dots$