Algebra Examples

$\frac{3 k^{2} - 12 k - 36}{4 k^{5} - 16 k^{3}} \div \frac{4 k - 24}{9 k^{4} - 18 k^{3}}$

Step 1

Set the denominator in $\frac{3 k^{2} - 12 k - 36}{4 k^{5} - 16 k^{3}}$ equal to $0$ to find where the expression is undefined.

$4 k^{5} - 16 k^{3} = 0$

Step 2

Solve for $k$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $4 k^{3}$ out of $4 k^{5} - 16 k^{3}$ .

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Step 2.1.1.1

Factor $4 k^{3}$ out of $4 k^{5}$ .

$4 k^{3} (k^{2}) - 16 k^{3} = 0$

Step 2.1.1.2

Factor $4 k^{3}$ out of $- 16 k^{3}$ .

$4 k^{3} (k^{2}) + 4 k^{3} (- 4) = 0$

Step 2.1.1.3

Factor $4 k^{3}$ out of $4 k^{3} (k^{2}) + 4 k^{3} (- 4)$ .

$4 k^{3} (k^{2} - 4) = 0$

Step 2.1.2

Rewrite $4$ as $2^{2}$ .

$4 k^{3} (k^{2} - 2^{2}) = 0$

Step 2.1.3

Factor.

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Step 2.1.3.1

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

$4 k^{3} ((k + 2) (k - 2)) = 0$

Step 2.1.3.2

Remove unnecessary parentheses.

$4 k^{3} (k + 2) (k - 2) = 0$

Step 2.2

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

$k^{3} = 0$

$k + 2 = 0$

$k - 2 = 0$

Step 2.3

Set $k^{3}$ equal to $0$ and solve for $k$ .

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Step 2.3.1

Set $k^{3}$ equal to $0$ .

$k^{3} = 0$

Step 2.3.2

Solve $k^{3} = 0$ for $k$ .

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Step 2.3.2.1

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

$k = \sqrt[3]{0}$

Step 2.3.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.3.2.2.1

Rewrite $0$ as $0^{3}$ .

$k = \sqrt[3]{0^{3}}$

Step 2.3.2.2.2

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

$k = 0$

Step 2.4

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

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Step 2.4.1

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

$k + 2 = 0$

Step 2.4.2

Subtract $2$ from both sides of the equation.

$k = - 2$

Step 2.5

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

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Step 2.5.1

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

$k - 2 = 0$

Step 2.5.2

Add $2$ to both sides of the equation.

$k = 2$

Step 2.6

The final solution is all the values that make $4 k^{3} (k + 2) (k - 2) = 0$ true.

$k = 0, - 2, 2$

Step 3

Set the denominator in $\frac{4 k - 24}{9 k^{4} - 18 k^{3}}$ equal to $0$ to find where the expression is undefined.

$9 k^{4} - 18 k^{3} = 0$

Step 4

Solve for $k$ .

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Step 4.1

Factor $9 k^{3}$ out of $9 k^{4} - 18 k^{3}$ .

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Step 4.1.1

Factor $9 k^{3}$ out of $9 k^{4}$ .

$9 k^{3} (k) - 18 k^{3} = 0$

Step 4.1.2

Factor $9 k^{3}$ out of $- 18 k^{3}$ .

$9 k^{3} (k) + 9 k^{3} (- 2) = 0$

Step 4.1.3

Factor $9 k^{3}$ out of $9 k^{3} (k) + 9 k^{3} (- 2)$ .

$9 k^{3} (k - 2) = 0$

Step 4.2

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

$k^{3} = 0$

$k - 2 = 0$

Step 4.3

Set $k^{3}$ equal to $0$ and solve for $k$ .

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Step 4.3.1

Set $k^{3}$ equal to $0$ .

$k^{3} = 0$

Step 4.3.2

Solve $k^{3} = 0$ for $k$ .

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Step 4.3.2.1

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

$k = \sqrt[3]{0}$

Step 4.3.2.2

Simplify $\sqrt[3]{0}$ .

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Step 4.3.2.2.1

Rewrite $0$ as $0^{3}$ .

$k = \sqrt[3]{0^{3}}$

Step 4.3.2.2.2

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

$k = 0$

Step 4.4

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

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Step 4.4.1

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

$k - 2 = 0$

Step 4.4.2

Add $2$ to both sides of the equation.

$k = 2$

Step 4.5

The final solution is all the values that make $9 k^{3} (k - 2) = 0$ true.

$k = 0, 2$

Step 5

Set the denominator in $\frac{3 k^{2} - 12 k - 36}{4 k^{5} - 16 k^{3}} \div \frac{4 k - 24}{9 k^{4} - 18 k^{3}}$ equal to $0$ to find where the expression is undefined.

$\frac{4 k - 24}{9 k^{4} - 18 k^{3}} = 0$

Step 6

Solve for $k$ .

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Step 6.1

Set the numerator equal to zero.

$4 k - 24 = 0$

Step 6.2

Solve the equation for $k$ .

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Step 6.2.1

Add $24$ to both sides of the equation.

$4 k = 24$

Step 6.2.2

Divide each term in $4 k = 24$ by $4$ and simplify.

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Step 6.2.2.1

Divide each term in $4 k = 24$ by $4$ .

$\frac{4 k}{4} = \frac{24}{4}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $4$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{4 k}{4} = \frac{24}{4}$

Step 6.2.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{24}{4}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Divide $24$ by $4$ .

$k = 6$

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 $0$ .

$k = - 2, k = 0, k = 2, k = 6$

Step 8