Find Where Undefined/Discontinuous (m^16+9m^8+8)/(4m^10+32m^2)

$\frac{m^{16} + 9 m^{8} + 8}{4 m^{10} + 32 m^{2}}$

Step 1

Set the denominator in $\frac{m^{16} + 9 m^{8} + 8}{4 m^{10} + 32 m^{2}}$ equal to $0$ to find where the expression is undefined.

$4 m^{10} + 32 m^{2} = 0$

Step 2

Solve for $m$ .

Factor $4 m^{2}$ out of $4 m^{10} + 32 m^{2}$ .

Factor $4 m^{2}$ out of $4 m^{10}$ .

$4 m^{2} (m^{8}) + 32 m^{2} = 0$

Factor $4 m^{2}$ out of $32 m^{2}$ .

$4 m^{2} (m^{8}) + 4 m^{2} (8) = 0$

Factor $4 m^{2}$ out of $4 m^{2} (m^{8}) + 4 m^{2} (8)$ .

$4 m^{2} (m^{8} + 8) = 0$

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

$m^{2} = 0$

$m^{8} + 8 = 0$

Set $m^{2}$ equal to $0$ and solve for $m$ .

Set $m^{2}$ equal to $0$ .

$m^{2} = 0$

Solve $m^{2} = 0$ for $m$ .

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

$m = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$m = \pm \sqrt{0^{2}}$

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

$m = \pm 0$

Plus or minus $0$ is $0$ .

$m = 0$

Set $m^{8} + 8$ equal to $0$ and solve for $m$ .

Set $m^{8} + 8$ equal to $0$ .

$m^{8} + 8 = 0$

Solve $m^{8} + 8 = 0$ for $m$ .

Subtract $8$ from both sides of the equation.

$m^{8} = - 8$

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

$m = \pm \sqrt[8]{- 8}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$m = \sqrt[8]{- 8}$

Next, use the negative value of the $\pm$ to find the second solution.

$m = - \sqrt[8]{- 8}$

The complete solution is the result of both the positive and negative portions of the solution.

$m = \sqrt[8]{- 8}, - \sqrt[8]{- 8}$

The final solution is all the values that make $4 m^{2} (m^{8} + 8) = 0$ true.

$m = 0, \sqrt[8]{- 8}, - \sqrt[8]{- 8}$

Step 3

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 4

Find Where Undefined/Discontinuous (m^16+9m^8+8)/(4m^10+32m^2)

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