Finite Math Examples

$\frac{\sqrt{16 m^{10}}}{4 m^{6}}$

Step 1

Set the denominator in $\frac{\sqrt{16 m^{10}}}{4 m^{6}}$ equal to $0$ to find where the expression is undefined.

$4 m^{6} = 0$

Step 2

Solve for $m$ .

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Step 2.1

Divide each term in $4 m^{6} = 0$ by $4$ and simplify.

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Step 2.1.1

Divide each term in $4 m^{6} = 0$ by $4$ .

$\frac{4 m^{6}}{4} = \frac{0}{4}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $4$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{4 m^{6}}{4} = \frac{0}{4}$

Step 2.1.2.1.2

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

$m^{6} = \frac{0}{4}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $4$ .

$m^{6} = 0$

Step 2.2

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

$m = \pm \sqrt[6]{0}$

Step 2.3

Simplify $\pm \sqrt[6]{0}$ .

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Step 2.3.1

Rewrite $0$ as $0^{6}$ .

$m = \pm \sqrt[6]{0^{6}}$

Step 2.3.2

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

$m = \pm 0$

Step 2.3.3

Plus or minus $0$ is $0$ .

$m = 0$

Step 3

Set the radicand in $\sqrt{16 m^{10}}$ less than $0$ to find where the expression is undefined.

$16 m^{10} < 0$

Step 4

Solve for $m$ .

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Step 4.1

Divide each term in $16 m^{10} < 0$ by $16$ and simplify.

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Step 4.1.1

Divide each term in $16 m^{10} < 0$ by $16$ .

$\frac{16 m^{10}}{16} < \frac{0}{16}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $16$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{16 m^{10}}{16} < \frac{0}{16}$

Step 4.1.2.1.2

Divide $m^{10}$ by $1$ .

$m^{10} < \frac{0}{16}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $16$ .

$m^{10} < 0$

Step 4.2

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

$\sqrt[10]{m^{10}} < \sqrt[10]{0}$

Step 4.3

Simplify the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Pull terms out from under the radical.

$| m | < \sqrt[10]{0}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Simplify $\sqrt[10]{0}$ .

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Step 4.3.2.1.1

Rewrite $0$ as $0^{10}$ .

$| m | < \sqrt[10]{0^{10}}$

Step 4.3.2.1.2

Pull terms out from under the radical.

$| m | < | 0 |$

Step 4.3.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| m | < 0$

Step 4.4

Write $| m | < 0$ as a piecewise.

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Step 4.4.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$m \geq 0$

Step 4.4.2

In the piece where $m$ is non-negative, remove the absolute value.

$m < 0$

Step 4.4.3

In the piece where $m$ is negative, remove the absolute value and multiply by $- 1$ .

$- m < 0$

Step 4.4.4

Write as a piecewise.

${\begin{cases} m < 0 & m \geq 0 \\ - m < 0 & m < 0 \end{cases}$

Step 4.5

Find the intersection of $m < 0$ and $m \geq 0$ .

No solution

Step 4.6

Solve $- m < 0$ when $m < 0$ .

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Step 4.6.1

Divide each term in $- m < 0$ by $- 1$ and simplify.

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Step 4.6.1.1

Divide each term in $- m < 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- m}{- 1} > \frac{0}{- 1}$

Step 4.6.1.2

Simplify the left side.

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Step 4.6.1.2.1

Dividing two negative values results in a positive value.

$\frac{m}{1} > \frac{0}{- 1}$

Step 4.6.1.2.2

Divide $m$ by $1$ .

$m > \frac{0}{- 1}$

Step 4.6.1.3

Simplify the right side.

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Step 4.6.1.3.1

Divide $0$ by $- 1$ .

$m > 0$

Step 4.6.2

Find the intersection of $m > 0$ and $m < 0$ .

No solution

Step 4.7

Find the union of the solutions.

No solution

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$m = 0$

Step 6