Basic Math Examples

$\frac{6}{a + 6} + \frac{6}{5 a} + \frac{36}{a^{2} + 6 a}$

Step 1

Factor $a$ out of $a^{2} + 6 a$ .

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Step 1.1

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

$\frac{6}{a + 6} + \frac{6}{5 a} + \frac{36}{a \cdot a + 6 a}$

Step 1.2

Factor $a$ out of $6 a$ .

$\frac{6}{a + 6} + \frac{6}{5 a} + \frac{36}{a \cdot a + a \cdot 6}$

Step 1.3

Factor $a$ out of $a \cdot a + a \cdot 6$ .

$\frac{6}{a + 6} + \frac{6}{5 a} + \frac{36}{a (a + 6)}$

Step 2

To write $\frac{6}{a + 6}$ as a fraction with a common denominator, multiply by $\frac{5 a}{5 a}$ .

$\frac{6}{a + 6} \cdot \frac{5 a}{5 a} + \frac{6}{5 a} + \frac{36}{a (a + 6)}$

Step 3

To write $\frac{6}{5 a}$ as a fraction with a common denominator, multiply by $\frac{a + 6}{a + 6}$ .

$\frac{6}{a + 6} \cdot \frac{5 a}{5 a} + \frac{6}{5 a} \cdot \frac{a + 6}{a + 6} + \frac{36}{a (a + 6)}$

Step 4

Write each expression with a common denominator of $(a + 6) \cdot 5 a$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{6}{a + 6}$ by $\frac{5 a}{5 a}$ .

$\frac{6 (5 a)}{(a + 6) (5 a)} + \frac{6}{5 a} \cdot \frac{a + 6}{a + 6} + \frac{36}{a (a + 6)}$

Step 4.2

Multiply $\frac{6}{5 a}$ by $\frac{a + 6}{a + 6}$ .

$\frac{6 (5 a)}{(a + 6) (5 a)} + \frac{6 (a + 6)}{5 a (a + 6)} + \frac{36}{a (a + 6)}$

Step 4.3

Reorder the factors of $(a + 6) (5 a)$ .

$\frac{6 (5 a)}{5 a (a + 6)} + \frac{6 (a + 6)}{5 a (a + 6)} + \frac{36}{a (a + 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{6 (5 a) + 6 (a + 6)}{5 a (a + 6)} + \frac{36}{a (a + 6)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $6$ out of $6 \cdot 5 a + 6 (a + 6)$ .

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Step 6.1.1

Factor $6$ out of $6 \cdot 5 a$ .

$\frac{6 (5 a) + 6 (a + 6)}{5 a (a + 6)} + \frac{36}{a (a + 6)}$

Step 6.1.2

Factor $6$ out of $6 (5 a) + 6 (a + 6)$ .

$\frac{6 (5 a + a + 6)}{5 a (a + 6)} + \frac{36}{a (a + 6)}$

Step 6.2

Add $5 a$ and $a$ .

$\frac{6 (6 a + 6)}{5 a (a + 6)} + \frac{36}{a (a + 6)}$

Step 6.3

Factor $6$ out of $6 a + 6$ .

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Step 6.3.1

Factor $6$ out of $6 a$ .

$\frac{6 (6 (a) + 6)}{5 a (a + 6)} + \frac{36}{a (a + 6)}$

Step 6.3.2

Factor $6$ out of $6$ .

$\frac{6 (6 (a) + 6 (1))}{5 a (a + 6)} + \frac{36}{a (a + 6)}$

Step 6.3.3

Factor $6$ out of $6 (a) + 6 (1)$ .

$\frac{6 (6 (a + 1))}{5 a (a + 6)} + \frac{36}{a (a + 6)}$

$\frac{6 \cdot 6 (a + 1)}{5 a (a + 6)} + \frac{36}{a (a + 6)}$

Step 6.4

Multiply $6$ by $6$ .

$\frac{36 (a + 1)}{5 a (a + 6)} + \frac{36}{a (a + 6)}$

Step 7

To write $\frac{36}{a (a + 6)}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{36 (a + 1)}{5 a (a + 6)} + \frac{36}{a (a + 6)} \cdot \frac{5}{5}$

Step 8

Write each expression with a common denominator of $5 a (a + 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{36}{a (a + 6)}$ by $\frac{5}{5}$ .

$\frac{36 (a + 1)}{5 a (a + 6)} + \frac{36 \cdot 5}{a (a + 6) \cdot 5}$

Step 8.2

Reorder the factors of $a (a + 6) \cdot 5$ .

$\frac{36 (a + 1)}{5 a (a + 6)} + \frac{36 \cdot 5}{5 a (a + 6)}$

Step 9

Combine the numerators over the common denominator.

$\frac{36 (a + 1) + 36 \cdot 5}{5 a (a + 6)}$

Step 10

Simplify the numerator.

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Step 10.1

Factor $36$ out of $36 (a + 1) + 36 \cdot 5$ .

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Step 10.1.1

Factor $36$ out of $36 \cdot 5$ .

$\frac{36 (a + 1) + 36 (5)}{5 a (a + 6)}$

Step 10.1.2

Factor $36$ out of $36 (a + 1) + 36 (5)$ .

$\frac{36 (a + 1 + 5)}{5 a (a + 6)}$

Step 10.2

Add $1$ and $5$ .

$\frac{36 (a + 6)}{5 a (a + 6)}$

Step 11

Cancel the common factor of $a + 6$ .

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Step 11.1

Cancel the common factor.

$\frac{36 (a + 6)}{5 a (a + 6)}$

Step 11.2

Rewrite the expression.

$\frac{36}{5 a}$