Algebra Examples

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

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.

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Step 2.1

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

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

Step 2.2

Divide $12$ by $2$ .

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

Step 2.3

Simplify the left side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

Multiply $6$ by $2$ .

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

Step 2.4

Simplify the right side.

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Step 2.4.1

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

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Step 2.4.1.1

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

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

Step 2.4.1.2

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

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

Step 2.4.1.3

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

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

Step 2.4.1.4

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

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Step 2.4.1.4.1

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

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

Step 2.4.1.4.2

Multiply $6$ by $2$ .

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

Step 3

Solve for $z$ .

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Step 3.1

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

$| z | = | z |$

Step 3.2

Solve for $z$ .

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Step 3.2.1

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

$z = z$

$z = - z$

$- z = z$

$- z = - z$

Step 3.2.2

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

$z = z$

$z = - z$

Step 3.2.3

Solve $z = z$ for $z$ .

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Step 3.2.3.1

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

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Step 3.2.3.1.1

Subtract $z$ from both sides of the equation.

$z - z = 0$

Step 3.2.3.1.2

Subtract $z$ from $z$ .

$0 = 0$

Step 3.2.3.2

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

All real numbers

Step 3.2.4

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

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Step 3.2.4.1

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

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Step 3.2.4.1.1

Add $z$ to both sides of the equation.

$z + z = 0$

Step 3.2.4.1.2

Add $z$ and $z$ .

$2 z = 0$

Step 3.2.4.2

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

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Step 3.2.4.2.1

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

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

Step 3.2.4.2.2

Simplify the left side.

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Step 3.2.4.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.4.2.2.1.1

Cancel the common factor.

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

Step 3.2.4.2.2.1.2

Divide $z$ by $1$ .

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

Step 3.2.4.2.3

Simplify the right side.

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Step 3.2.4.2.3.1

Divide $0$ by $2$ .

$z = 0$

Step 3.2.5

List all of the solutions.

$z = 0$