Algebra Examples

\sqrt{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}

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

Step 2

Simplify each side of the equation.

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Step 2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{z^{12}}

$\sqrt{z^{12}}$ as

z^{\frac{12}{2}}

$z^{\frac{12}{2}}$ .

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

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

Step 2.2

Divide

12

$12$ by

2

$2$ .

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

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

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Step 2.3.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.3.1.2

Multiply

6

$6$ by

2

$2$ .

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

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

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

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

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

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

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Step 2.4.1.1

Apply the product rule to

- z^{6}

$- z^{6}$ .

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

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

Step 2.4.1.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 2.4.1.3

Multiply

{(z^{6})}^{2}

${(z^{6})}^{2}$ by

1

$1$ .

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

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

Step 2.4.1.4

Multiply the exponents in

{(z^{6})}^{2}

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

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Step 2.4.1.4.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.4.1.4.2

Multiply

6

$6$ by

2

$2$ .

z^{12} = z^{12}

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

z^{12} = z^{12}

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

z^{12} = z^{12}

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

z^{12} = z^{12}

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

z^{12} = z^{12}

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

Step 3

Solve for

z

$z$ .

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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 |

$| z | = | z |$

Step 3.2

Solve for

z

$z$ .

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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

$z = - z$

- 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

$z = z$

z = - z

$z = - z$

Step 3.2.3

Solve

z = z

$z = z$ for

z

$z$ .

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Step 3.2.3.1

Move all terms containing

z

$z$ to the left side of the equation.

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Step 3.2.3.1.1

Subtract

z

$z$ from both sides of the equation.

z - z = 0

$z - z = 0$

Step 3.2.3.1.2

Subtract

z

$z$ from

z

$z$ .

0 = 0

$0 = 0$

0 = 0

$0 = 0$

Step 3.2.3.2

Since

0 = 0

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

All real numbers

Step 3.2.4

Solve

z = - z

$z = - z$ for

z

$z$ .

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Step 3.2.4.1

Move all terms containing

z

$z$ to the left side of the equation.

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Step 3.2.4.1.1

Add

z

$z$ to both sides of the equation.

z + z = 0

$z + z = 0$

Step 3.2.4.1.2

Add

z

$z$ and

z

$z$ .

2 z = 0

$2 z = 0$

2 z = 0

$2 z = 0$

Step 3.2.4.2

Divide each term in

2 z = 0

$2 z = 0$ by

2

$2$ and simplify.

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Step 3.2.4.2.1

Divide each term in

2 z = 0

$2 z = 0$ by

2

$2$ .

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

$2$ .

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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.

$z = 0$