Find Where Undefined/Discontinuous ((k^4-k^2)/(18k^6))÷((k^2-37k-38)/(12k^2-24k))

$\frac{k^{4} - k^{2}}{18 k^{6}} \div \frac{k^{2} - 37 k - 38}{12 k^{2} - 24 k}$

Step 1

Set the denominator in $\frac{k^{4} - k^{2}}{18 k^{6}}$ equal to $0$ to find where the expression is undefined.

$18 k^{6} = 0$

Step 2

Solve for $k$ .

Divide each term in $18 k^{6} = 0$ by $18$ and simplify.

Divide each term in $18 k^{6} = 0$ by $18$ .

$\frac{18 k^{6}}{18} = \frac{0}{18}$

Simplify the left side.

Cancel the common factor of $18$ .

Cancel the common factor.

$\frac{18 k^{6}}{18} = \frac{0}{18}$

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

$k^{6} = \frac{0}{18}$

Simplify the right side.

Divide $0$ by $18$ .

$k^{6} = 0$

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

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

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

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

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

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

$k = \pm 0$

Plus or minus $0$ is $0$ .

$k = 0$

Step 3

Set the denominator in $\frac{k^{2} - 37 k - 38}{12 k^{2} - 24 k}$ equal to $0$ to find where the expression is undefined.

$12 k^{2} - 24 k = 0$

Step 4

Solve for $k$ .

Factor $12 k$ out of $12 k^{2} - 24 k$ .

Factor $12 k$ out of $12 k^{2}$ .

$12 k (k) - 24 k = 0$

Factor $12 k$ out of $- 24 k$ .

$12 k (k) + 12 k (- 2) = 0$

Factor $12 k$ out of $12 k (k) + 12 k (- 2)$ .

$12 k (k - 2) = 0$

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

$k = 0$

$k - 2 = 0$

Set $k$ equal to $0$ .

$k = 0$

Set $k - 2$ equal to $0$ and solve for $k$ .

Set $k - 2$ equal to $0$ .

$k - 2 = 0$

Add $2$ to both sides of the equation.

$k = 2$

The final solution is all the values that make $12 k (k - 2) = 0$ true.

$k = 0, 2$

Step 5

Set the denominator in $\frac{\frac{k^{4} - k^{2}}{18 k^{6}}}{\frac{k^{2} - 37 k - 38}{12 k^{2} - 24 k}}$ equal to $0$ to find where the expression is undefined.

$\frac{k^{2} - 37 k - 38}{12 k^{2} - 24 k} = 0$

Step 6

Solve for $k$ .

Set the numerator equal to zero.

$k^{2} - 37 k - 38 = 0$

Solve the equation for $k$ .

Factor $k^{2} - 37 k - 38$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 38$ and whose sum is $- 37$ .

$- 38, 1$

Write the factored form using these integers.

$(k - 38) (k + 1) = 0$

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

$k - 38 = 0$

$k + 1 = 0$

Set $k - 38$ equal to $0$ and solve for $k$ .

Set $k - 38$ equal to $0$ .

$k - 38 = 0$

Add $38$ to both sides of the equation.

$k = 38$

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

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

$k + 1 = 0$

Subtract $1$ from both sides of the equation.

$k = - 1$

The final solution is all the values that make $(k - 38) (k + 1) = 0$ true.

$k = 38, - 1$

Step 7

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 $0$ .

$k = - 1, k = 0, k = 2, k = 38$

Step 8

Find Where Undefined/Discontinuous ((k^4-k^2)/(18k^6))÷((k^2-37k-38)/(12k^2-24k))

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