Solve for z square root of z^12=-z^6

$\sqrt{z^{12}} = - z^{6}$

Step 1

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

${\sqrt{z^{12}}}^{2} = {(- z^{6})}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z^{12}}$ as $z^{\frac{12}{2}}$ .

${(z^{\frac{12}{2}})}^{2} = {(- z^{6})}^{2}$

Divide $12$ by $2$ .

${(z^{6})}^{2} = {(- z^{6})}^{2}$

Simplify the left side.

Multiply the exponents in ${(z^{6})}^{2}$ .

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

$z^{6 \cdot 2} = {(- z^{6})}^{2}$

Multiply $6$ by $2$ .

$z^{12} = {(- z^{6})}^{2}$

Simplify the right side.

Simplify ${(- z^{6})}^{2}$ .

Apply the product rule to $- z^{6}$ .

$z^{12} = {(- 1)}^{2} {(z^{6})}^{2}$

Raise $- 1$ to the power of $2$ .

$z^{12} = 1 {(z^{6})}^{2}$

Multiply ${(z^{6})}^{2}$ by $1$ .

$z^{12} = {(z^{6})}^{2}$

Multiply the exponents in ${(z^{6})}^{2}$ .

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

$z^{12} = z^{6 \cdot 2}$

Multiply $6$ by $2$ .

$z^{12} = z^{12}$

Step 3

Solve for $z$ .

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| z | = | z |$

Solve for $z$ .

Rewrite the absolute value equation as four equations without absolute value bars.

$z = z$

$z = - z$

$- z = z$

$- z = - z$

After simplifying, there are only two unique equations to be solved.

$z = z$

$z = - z$

Solve $z = z$ for $z$ .

Move all terms containing $z$ to the left side of the equation.

Subtract $z$ from both sides of the equation.

$z - z = 0$

Subtract $z$ from $z$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true.

All real numbers

Solve $z = - z$ for $z$ .

Move all terms containing $z$ to the left side of the equation.

Add $z$ to both sides of the equation.

$z + z = 0$

Add $z$ and $z$ .

$2 z = 0$

Divide each term in $2 z = 0$ by $2$ and simplify.

Divide each term in $2 z = 0$ by $2$ .

$\frac{2 z}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 z}{2} = \frac{0}{2}$

Divide $z$ by $1$ .

$z = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$z = 0$

List all of the solutions.

$z = 0$

Solve for z square root of z^12=-z^6

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