Basic Math Examples

$1 + \frac{4}{n} + \frac{3}{n^{2}} \div 1 + \frac{10}{n} + \frac{21}{n^{2}}$

Step 1

Combine the numerators over the common denominator.

$1 + \frac{4 + 10}{n} + \frac{\frac{3}{n^{2}}}{1} + \frac{21}{n^{2}}$

Step 2

Simplify the expression.

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Step 2.1

Add $4$ and $10$ .

$1 + \frac{14}{n} + \frac{\frac{3}{n^{2}}}{1} + \frac{21}{n^{2}}$

Step 2.2

Divide $\frac{3}{n^{2}}$ by $1$ .

$1 + \frac{14}{n} + \frac{3}{n^{2}} + \frac{21}{n^{2}}$

Step 3

Combine the numerators over the common denominator.

$1 + \frac{14}{n} + \frac{3 + 21}{n^{2}}$

Step 4

Simplify the expression.

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Step 4.1

Add $3$ and $21$ .

$1 + \frac{14}{n} + \frac{24}{n^{2}}$

Step 4.2

Write $1$ as a fraction with a common denominator.

$\frac{n}{n} + \frac{14}{n} + \frac{24}{n^{2}}$

Step 5

Combine the numerators over the common denominator.

$\frac{n + 14}{n} + \frac{24}{n^{2}}$

Step 6

To write $\frac{n + 14}{n}$ as a fraction with a common denominator, multiply by $\frac{n}{n}$ .

$\frac{n + 14}{n} \cdot \frac{n}{n} + \frac{24}{n^{2}}$

Step 7

Write each expression with a common denominator of $n^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{n + 14}{n}$ by $\frac{n}{n}$ .

$\frac{(n + 14) n}{n \cdot n} + \frac{24}{n^{2}}$

Step 7.2

Multiply $n$ by $n$ .

$\frac{(n + 14) n}{n^{2}} + \frac{24}{n^{2}}$

Step 8

Combine the numerators over the common denominator.

$\frac{(n + 14) n + 24}{n^{2}}$

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

$\frac{n \cdot n + 14 n + 24}{n^{2}}$

Step 9.2

Multiply $n$ by $n$ .

$\frac{n^{2} + 14 n + 24}{n^{2}}$

Step 9.3

Factor $n^{2} + 14 n + 24$ using the AC method.

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Step 9.3.1

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 $24$ and whose sum is $14$ .

$2, 12$

Step 9.3.2

Write the factored form using these integers.

$\frac{(n + 2) (n + 12)}{n^{2}}$