Solve for z 4 = square root of z^2-9

$4 = \sqrt{z^{2} - 9}$

Rewrite the equation as $\sqrt{z^{2} - 9} = 4$ .

$\sqrt{z^{2} - 9} = 4$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{z^{2} - 9}}^{2} = 4^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z^{2} - 9}$ as ${(z^{2} - 9)}^{\frac{1}{2}}$ .

${({(z^{2} - 9)}^{\frac{1}{2}})}^{2} = 4^{2}$

Simplify the left side.

Simplify ${({(z^{2} - 9)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(z^{2} - 9)}^{\frac{1}{2}})}^{2}$ .

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

${(z^{2} - 9)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(z^{2} - 9)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Rewrite the expression.

${(z^{2} - 9)}^{1} = 4^{2}$

Simplify.

$z^{2} - 9 = 4^{2}$

Simplify the right side.

Raise $4$ to the power of $2$ .

$z^{2} - 9 = 16$

Solve for $z$ .

Move all terms not containing $z$ to the right side of the equation.

Add $9$ to both sides of the equation.

$z^{2} = 16 + 9$

Add $16$ and $9$ .

$z^{2} = 25$

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

$z = \pm \sqrt{25}$

Simplify $\pm \sqrt{25}$ .

Rewrite $25$ as $5^{2}$ .

$z = \pm \sqrt{5^{2}}$

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

$z = \pm 5$

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.

$z = 5$

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

$z = - 5$

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

$z = 5, - 5$