Basic Math Examples

$\frac{z^{8}}{z^{2}} = z^{4}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Reduce the expression $\frac{z^{8}}{z^{2}}$ by cancelling the common factors.

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Step 1.1.1

Factor $z^{2}$ out of $z^{8}$ .

$\frac{z^{2} z^{6}}{z^{2}} = z^{4}$

Step 1.1.2

Multiply by $1$ .

$\frac{z^{2} z^{6}}{z^{2} \cdot 1} = z^{4}$

Step 1.1.3

Cancel the common factor.

$\frac{z^{2} z^{6}}{z^{2} \cdot 1} = z^{4}$

Step 1.1.4

Rewrite the expression.

$\frac{z^{6}}{1} = z^{4}$

Step 1.2

Divide $z^{6}$ by $1$ .

$z^{6} = z^{4}$

Step 2

Solve the equation.

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Step 2.1

Subtract $z^{4}$ from both sides of the equation.

$z^{6} - z^{4} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $z^{4}$ out of $z^{6} - z^{4}$ .

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Step 2.2.1.1

Factor $z^{4}$ out of $z^{6}$ .

$z^{4} z^{2} - z^{4} = 0$

Step 2.2.1.2

Factor $z^{4}$ out of $- z^{4}$ .

$z^{4} z^{2} + z^{4} \cdot - 1 = 0$

Step 2.2.1.3

Factor $z^{4}$ out of $z^{4} z^{2} + z^{4} \cdot - 1$ .

$z^{4} (z^{2} - 1) = 0$

Step 2.2.2

Rewrite $1$ as $1^{2}$ .

$z^{4} (z^{2} - 1^{2}) = 0$

Step 2.2.3

Factor.

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Step 2.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 1$ .

$z^{4} ((z + 1) (z - 1)) = 0$

Step 2.2.3.2

Remove unnecessary parentheses.

$z^{4} (z + 1) (z - 1) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$z^{4} = 0$

$z + 1 = 0$

$z - 1 = 0$

Step 2.4

Set $z^{4}$ equal to $0$ and solve for $z$ .

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Step 2.4.1

Set $z^{4}$ equal to $0$ .

$z^{4} = 0$

Step 2.4.2

Solve $z^{4} = 0$ for $z$ .

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Step 2.4.2.1

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

$z = \pm \sqrt[4]{0}$

Step 2.4.2.2

Simplify $\pm \sqrt[4]{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{4}$ .

$z = \pm \sqrt[4]{0^{4}}$

Step 2.4.2.2.2

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

$z = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$z = 0$

Step 2.5

Set $z + 1$ equal to $0$ and solve for $z$ .

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Step 2.5.1

Set $z + 1$ equal to $0$ .

$z + 1 = 0$

Step 2.5.2

Subtract $1$ from both sides of the equation.

$z = - 1$

Step 2.6

Set $z - 1$ equal to $0$ and solve for $z$ .

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Step 2.6.1

Set $z - 1$ equal to $0$ .

$z - 1 = 0$

Step 2.6.2

Add $1$ to both sides of the equation.

$z = 1$

Step 2.7

The final solution is all the values that make $z^{4} (z + 1) (z - 1) = 0$ true.

$z = 0, - 1, 1$

Step 3

Exclude the solutions that do not make $\frac{z^{8}}{z^{2}} = z^{4}$ true.

$z = - 1, 1$