Solve for z square root of (z+1)^2=2

$\sqrt{{(z + 1)}^{2}} = 2$

Step 1

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

${\sqrt{{(z + 1)}^{2}}}^{2} = 2^{2}$

Step 2

Simplify each side of the equation.

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

${({(z + 1)}^{\frac{2}{2}})}^{2} = 2^{2}$

Divide $2$ by $2$ .

${({(z + 1)}^{1})}^{2} = 2^{2}$

Simplify the left side.

Multiply the exponents in ${({(z + 1)}^{1})}^{2}$ .

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

${(z + 1)}^{1 \cdot 2} = 2^{2}$

Multiply $2$ by $1$ .

${(z + 1)}^{2} = 2^{2}$

Simplify the right side.

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

${(z + 1)}^{2} = 4$

Step 3

Solve for $z$ .

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

$z + 1 = \pm \sqrt{4}$

Simplify $\pm \sqrt{4}$ .

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

$z + 1 = \pm \sqrt{2^{2}}$

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

$z + 1 = \pm 2$

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.

$z + 1 = 2$

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

Subtract $1$ from both sides of the equation.

$z = 2 - 1$

Subtract $1$ from $2$ .

$z = 1$

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

$z + 1 = - 2$

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

Subtract $1$ from both sides of the equation.

$z = - 2 - 1$

Subtract $1$ from $- 2$ .

$z = - 3$

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

$z = 1, - 3$