Basic Math Examples

$\frac{2 n}{n^{3} - 5 n^{2}} + \frac{2}{n^{2} + 5 n}$

Step 1

Simplify each term.

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Step 1.1

Factor $n^{2}$ out of $n^{3} - 5 n^{2}$ .

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Step 1.1.1

Factor $n^{2}$ out of $n^{3}$ .

$\frac{2 n}{n^{2} n - 5 n^{2}} + \frac{2}{n^{2} + 5 n}$

Step 1.1.2

Factor $n^{2}$ out of $- 5 n^{2}$ .

$\frac{2 n}{n^{2} n + n^{2} \cdot - 5} + \frac{2}{n^{2} + 5 n}$

Step 1.1.3

Factor $n^{2}$ out of $n^{2} n + n^{2} \cdot - 5$ .

$\frac{2 n}{n^{2} (n - 5)} + \frac{2}{n^{2} + 5 n}$

Step 1.2

Cancel the common factor of $n$ and $n^{2}$ .

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Step 1.2.1

Factor $n$ out of $2 n$ .

$\frac{n \cdot 2}{n^{2} (n - 5)} + \frac{2}{n^{2} + 5 n}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $n$ out of $n^{2} (n - 5)$ .

$\frac{n \cdot 2}{n (n (n - 5))} + \frac{2}{n^{2} + 5 n}$

Step 1.2.2.2

Cancel the common factor.

$\frac{n \cdot 2}{n (n (n - 5))} + \frac{2}{n^{2} + 5 n}$

Step 1.2.2.3

Rewrite the expression.

$\frac{2}{n (n - 5)} + \frac{2}{n^{2} + 5 n}$

Step 1.3

Factor $n$ out of $n^{2} + 5 n$ .

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Step 1.3.1

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

$\frac{2}{n (n - 5)} + \frac{2}{n \cdot n + 5 n}$

Step 1.3.2

Factor $n$ out of $5 n$ .

$\frac{2}{n (n - 5)} + \frac{2}{n \cdot n + n \cdot 5}$

Step 1.3.3

Factor $n$ out of $n \cdot n + n \cdot 5$ .

$\frac{2}{n (n - 5)} + \frac{2}{n (n + 5)}$

Step 2

To write $\frac{2}{n (n - 5)}$ as a fraction with a common denominator, multiply by $\frac{n + 5}{n + 5}$ .

$\frac{2}{n (n - 5)} \cdot \frac{n + 5}{n + 5} + \frac{2}{n (n + 5)}$

Step 3

To write $\frac{2}{n (n + 5)}$ as a fraction with a common denominator, multiply by $\frac{n - 5}{n - 5}$ .

$\frac{2}{n (n - 5)} \cdot \frac{n + 5}{n + 5} + \frac{2}{n (n + 5)} \cdot \frac{n - 5}{n - 5}$

Step 4

Write each expression with a common denominator of $n (n - 5) (n + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{n (n - 5)}$ by $\frac{n + 5}{n + 5}$ .

$\frac{2 (n + 5)}{n (n - 5) (n + 5)} + \frac{2}{n (n + 5)} \cdot \frac{n - 5}{n - 5}$

Step 4.2

Multiply $\frac{2}{n (n + 5)}$ by $\frac{n - 5}{n - 5}$ .

$\frac{2 (n + 5)}{n (n - 5) (n + 5)} + \frac{2 (n - 5)}{n (n + 5) (n - 5)}$

Step 4.3

Reorder the factors of $n (n - 5) (n + 5)$ .

$\frac{2 (n + 5)}{n (n + 5) (n - 5)} + \frac{2 (n - 5)}{n (n + 5) (n - 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 (n + 5) + 2 (n - 5)}{n (n + 5) (n - 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $2$ out of $2 (n + 5) + 2 (n - 5)$ .

$\frac{2 (n + 5 + n - 5)}{n (n + 5) (n - 5)}$

Step 6.2

Add $n$ and $n$ .

$\frac{2 (2 n + 5 - 5)}{n (n + 5) (n - 5)}$

Step 6.3

Subtract $5$ from $5$ .

$\frac{2 (2 n + 0)}{n (n + 5) (n - 5)}$

Step 6.4

Add $2 n$ and $0$ .

$\frac{2 \cdot 2 n}{n (n + 5) (n - 5)}$

Step 6.5

Multiply $2$ by $2$ .

$\frac{4 n}{n (n + 5) (n - 5)}$

Step 7

Cancel the common factor of $n$ .

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Step 7.1

Cancel the common factor.

$\frac{4 n}{n (n + 5) (n - 5)}$

Step 7.2

Rewrite the expression.

$\frac{4}{(n + 5) (n - 5)}$