Solve Using the Square Root Property z^2=-81

$z^{2} = - 81$

Step 1

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

$z = \pm \sqrt{- 81}$

Step 2

Simplify $\pm \sqrt{- 81}$ .

Rewrite $- 81$ as $- 1 (81)$ .

$z = \pm \sqrt{- 1 (81)}$

Rewrite $\sqrt{- 1 (81)}$ as $\sqrt{- 1} \cdot \sqrt{81}$ .

$z = \pm \sqrt{- 1} \cdot \sqrt{81}$

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

$z = \pm i \cdot \sqrt{81}$

Rewrite $81$ as $9^{2}$ .

$z = \pm i \cdot \sqrt{9^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$z = \pm i \cdot 9$

Move $9$ to the left of $i$ .

$z = \pm 9 i$

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.

$z = 9 i$

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

$z = - 9 i$

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

$z = 9 i, - 9 i$