Algebra Examples

$\frac{12 n^{7} + 144 n^{6}}{n^{3} + 8 n^{2} - 48 n} \cdot \frac{n^{2} - 8 n + 16}{n^{2}}$

Step 1

Set the denominator in $\frac{12 n^{7} + 144 n^{6}}{n^{3} + 8 n^{2} - 48 n}$ equal to $0$ to find where the expression is undefined.

$n^{3} + 8 n^{2} - 48 n = 0$

Step 2

Solve for $n$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $n$ out of $n^{3} + 8 n^{2} - 48 n$ .

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Step 2.1.1.1

Factor $n$ out of $n^{3}$ .

$n \cdot n^{2} + 8 n^{2} - 48 n = 0$

Step 2.1.1.2

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

$n \cdot n^{2} + n (8 n) - 48 n = 0$

Step 2.1.1.3

Factor $n$ out of $- 48 n$ .

$n \cdot n^{2} + n (8 n) + n \cdot - 48 = 0$

Step 2.1.1.4

Factor $n$ out of $n \cdot n^{2} + n (8 n)$ .

$n (n^{2} + 8 n) + n \cdot - 48 = 0$

Step 2.1.1.5

Factor $n$ out of $n (n^{2} + 8 n) + n \cdot - 48$ .

$n (n^{2} + 8 n - 48) = 0$

Step 2.1.2

Factor.

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Step 2.1.2.1

Factor $n^{2} + 8 n - 48$ using the AC method.

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Step 2.1.2.1.1

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 $- 48$ and whose sum is $8$ .

$- 4, 12$

Step 2.1.2.1.2

Write the factored form using these integers.

$n ((n - 4) (n + 12)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$n (n - 4) (n + 12) = 0$

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 $0$ .

$n = 0$

$n - 4 = 0$

$n + 12 = 0$

Step 2.3

Set $n$ equal to $0$ .

$n = 0$

Step 2.4

Set $n - 4$ equal to $0$ and solve for $n$ .

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Step 2.4.1

Set $n - 4$ equal to $0$ .

$n - 4 = 0$

Step 2.4.2

Add $4$ to both sides of the equation.

$n = 4$

Step 2.5

Set $n + 12$ equal to $0$ and solve for $n$ .

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Step 2.5.1

Set $n + 12$ equal to $0$ .

$n + 12 = 0$

Step 2.5.2

Subtract $12$ from both sides of the equation.

$n = - 12$

Step 2.6

The final solution is all the values that make $n (n - 4) (n + 12) = 0$ true.

$n = 0, 4, - 12$

Step 3

Set the denominator in $\frac{n^{2} - 8 n + 16}{n^{2}}$ equal to $0$ to find where the expression is undefined.

$n^{2} = 0$

Step 4

Solve for $n$ .

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Step 4.1

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

$n = \pm \sqrt{0}$

Step 4.2

Simplify $\pm \sqrt{0}$ .

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Step 4.2.1

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

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

Step 4.2.2

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

$n = \pm 0$

Step 4.2.3

Plus or minus $0$ is $0$ .

$n = 0$

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 $0$ .

$n = - 12, n = 0, n = 4$

Step 6