Solve Using the Square Root Property (8k-1)^2=-27

${(8 k - 1)}^{2} = - 27$

Step 1

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

$8 k - 1 = \pm \sqrt{- 27}$

Step 2

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

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

$8 k - 1 = \pm \sqrt{- 1 (27)}$

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

$8 k - 1 = \pm \sqrt{- 1} \cdot \sqrt{27}$

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

$8 k - 1 = \pm i \cdot \sqrt{27}$

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

Factor $9$ out of $27$ .

$8 k - 1 = \pm i \cdot \sqrt{9 (3)}$

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

$8 k - 1 = \pm i \cdot \sqrt{3^{2} \cdot 3}$

Pull terms out from under the radical.

$8 k - 1 = \pm i \cdot (3 \sqrt{3})$

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

$8 k - 1 = \pm 3 i \sqrt{3}$

Step 3

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

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

$8 k - 1 = 3 i \sqrt{3}$

Add $1$ to both sides of the equation.

$8 k = 3 i \sqrt{3} + 1$

Divide each term in $8 k = 3 i \sqrt{3} + 1$ by $8$ and simplify.

Divide each term in $8 k = 3 i \sqrt{3} + 1$ by $8$ .

$\frac{8 k}{8} = \frac{3 i \sqrt{3}}{8} + \frac{1}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 k}{8} = \frac{3 i \sqrt{3}}{8} + \frac{1}{8}$

Divide $k$ by $1$ .

$k = \frac{3 i \sqrt{3}}{8} + \frac{1}{8}$

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

$8 k - 1 = - 3 i \sqrt{3}$

Add $1$ to both sides of the equation.

$8 k = - 3 i \sqrt{3} + 1$

Divide each term in $8 k = - 3 i \sqrt{3} + 1$ by $8$ and simplify.

Divide each term in $8 k = - 3 i \sqrt{3} + 1$ by $8$ .

$\frac{8 k}{8} = \frac{- 3 i \sqrt{3}}{8} + \frac{1}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 k}{8} = \frac{- 3 i \sqrt{3}}{8} + \frac{1}{8}$

Divide $k$ by $1$ .

$k = \frac{- 3 i \sqrt{3}}{8} + \frac{1}{8}$

Simplify the right side.

Move the negative in front of the fraction.

$k = - \frac{3 i \sqrt{3}}{8} + \frac{1}{8}$

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

$k = \frac{3 i \sqrt{3}}{8} + \frac{1}{8}, - \frac{3 i \sqrt{3}}{8} + \frac{1}{8}$

Solve Using the Square Root Property (8k-1)^2=-27

Related Questions

Support LCMGCF.com by disabling your adblocker.