Solve Using the Square Root Property (7k-1)^2=-8

${(7 k - 1)}^{2} = - 8$

Step 1

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

$7 k - 1 = \pm \sqrt{- 8}$

Step 2

Simplify $\pm \sqrt{- 8}$ .

Rewrite $- 8$ as $- 1 (8)$ .

$7 k - 1 = \pm \sqrt{- 1 (8)}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$7 k - 1 = \pm \sqrt{- 1} \cdot \sqrt{8}$

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

$7 k - 1 = \pm i \cdot \sqrt{8}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$7 k - 1 = \pm i \cdot \sqrt{4 (2)}$

Rewrite $4$ as $2^{2}$ .

$7 k - 1 = \pm i \cdot \sqrt{2^{2} \cdot 2}$

Pull terms out from under the radical.

$7 k - 1 = \pm i \cdot (2 \sqrt{2})$

Move $2$ to the left of $i$ .

$7 k - 1 = \pm 2 i \sqrt{2}$

Step 3

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

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

$7 k - 1 = 2 i \sqrt{2}$

Add $1$ to both sides of the equation.

$7 k = 2 i \sqrt{2} + 1$

Divide each term in $7 k = 2 i \sqrt{2} + 1$ by $7$ and simplify.

Divide each term in $7 k = 2 i \sqrt{2} + 1$ by $7$ .

$\frac{7 k}{7} = \frac{2 i \sqrt{2}}{7} + \frac{1}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 k}{7} = \frac{2 i \sqrt{2}}{7} + \frac{1}{7}$

Divide $k$ by $1$ .

$k = \frac{2 i \sqrt{2}}{7} + \frac{1}{7}$

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

$7 k - 1 = - 2 i \sqrt{2}$

Add $1$ to both sides of the equation.

$7 k = - 2 i \sqrt{2} + 1$

Divide each term in $7 k = - 2 i \sqrt{2} + 1$ by $7$ and simplify.

Divide each term in $7 k = - 2 i \sqrt{2} + 1$ by $7$ .

$\frac{7 k}{7} = \frac{- 2 i \sqrt{2}}{7} + \frac{1}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 k}{7} = \frac{- 2 i \sqrt{2}}{7} + \frac{1}{7}$

Divide $k$ by $1$ .

$k = \frac{- 2 i \sqrt{2}}{7} + \frac{1}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$k = - \frac{2 i \sqrt{2}}{7} + \frac{1}{7}$

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

$k = \frac{2 i \sqrt{2}}{7} + \frac{1}{7}, - \frac{2 i \sqrt{2}}{7} + \frac{1}{7}$

Solve Using the Square Root Property (7k-1)^2=-8

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