Finite Math Examples

${(6 k - 1)}^{2} = - 27$

Step 1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$6 k - 1 = \pm \sqrt{- 27}$

Step 2

Simplify $\pm \sqrt{- 27}$ .

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Step 2.1

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

$6 k - 1 = \pm \sqrt{- 1 (27)}$

Step 2.2

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

$6 k - 1 = \pm \sqrt{- 1} \cdot \sqrt{27}$

Step 2.3

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

$6 k - 1 = \pm i \cdot \sqrt{27}$

Step 2.4

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

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Step 2.4.1

Factor $9$ out of $27$ .

$6 k - 1 = \pm i \cdot \sqrt{9 (3)}$

Step 2.4.2

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

$6 k - 1 = \pm i \cdot \sqrt{3^{2} \cdot 3}$

Step 2.5

Pull terms out from under the radical.

$6 k - 1 = \pm i \cdot (3 \sqrt{3})$

Step 2.6

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

$6 k - 1 = \pm 3 i \sqrt{3}$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$6 k - 1 = 3 i \sqrt{3}$

Step 3.2

Add $1$ to both sides of the equation.

$6 k = 3 i \sqrt{3} + 1$

Step 3.3

Divide each term in $6 k = 3 i \sqrt{3} + 1$ by $6$ and simplify.

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Step 3.3.1

Divide each term in $6 k = 3 i \sqrt{3} + 1$ by $6$ .

$\frac{6 k}{6} = \frac{3 i \sqrt{3}}{6} + \frac{1}{6}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $6$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{6 k}{6} = \frac{3 i \sqrt{3}}{6} + \frac{1}{6}$

Step 3.3.2.1.2

Divide $k$ by $1$ .

$k = \frac{3 i \sqrt{3}}{6} + \frac{1}{6}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $3$ and $6$ .

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Step 3.3.3.1.1

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

$k = \frac{3 (i \sqrt{3})}{6} + \frac{1}{6}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $3$ out of $6$ .

$k = \frac{3 (i \sqrt{3})}{3 \cdot 2} + \frac{1}{6}$

Step 3.3.3.1.2.2

Cancel the common factor.

$k = \frac{3 (i \sqrt{3})}{3 \cdot 2} + \frac{1}{6}$

Step 3.3.3.1.2.3

Rewrite the expression.

$k = \frac{i \sqrt{3}}{2} + \frac{1}{6}$

Step 3.4

Next, use the negative value of the $\pm$ to find the second solution.

$6 k - 1 = - 3 i \sqrt{3}$

Step 3.5

Add $1$ to both sides of the equation.

$6 k = - 3 i \sqrt{3} + 1$

Step 3.6

Divide each term in $6 k = - 3 i \sqrt{3} + 1$ by $6$ and simplify.

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Step 3.6.1

Divide each term in $6 k = - 3 i \sqrt{3} + 1$ by $6$ .

$\frac{6 k}{6} = \frac{- 3 i \sqrt{3}}{6} + \frac{1}{6}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $6$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{6 k}{6} = \frac{- 3 i \sqrt{3}}{6} + \frac{1}{6}$

Step 3.6.2.1.2

Divide $k$ by $1$ .

$k = \frac{- 3 i \sqrt{3}}{6} + \frac{1}{6}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Simplify each term.

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Step 3.6.3.1.1

Cancel the common factor of $- 3$ and $6$ .

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Step 3.6.3.1.1.1

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

$k = \frac{3 (- i \sqrt{3})}{6} + \frac{1}{6}$

Step 3.6.3.1.1.2

Cancel the common factors.

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Step 3.6.3.1.1.2.1

Factor $3$ out of $6$ .

$k = \frac{3 (- i \sqrt{3})}{3 (2)} + \frac{1}{6}$

Step 3.6.3.1.1.2.2

Cancel the common factor.

$k = \frac{3 (- i \sqrt{3})}{3 \cdot 2} + \frac{1}{6}$

Step 3.6.3.1.1.2.3

Rewrite the expression.

$k = \frac{- i \sqrt{3}}{2} + \frac{1}{6}$

Step 3.6.3.1.2

Move the negative in front of the fraction.

$k = - \frac{i \sqrt{3}}{2} + \frac{1}{6}$

Step 3.7

The complete solution is the result of both the positive and negative portions of the solution.

$k = \frac{i \sqrt{3}}{2} + \frac{1}{6}, - \frac{i \sqrt{3}}{2} + \frac{1}{6}$