Basic Math Examples

$\frac{w + 6}{w^{2}} + \frac{w + 9}{6 w}$

Step 1

To write $\frac{w + 6}{w^{2}}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{w + 6}{w^{2}} \cdot \frac{6}{6} + \frac{w + 9}{6 w}$

Step 2

To write $\frac{w + 9}{6 w}$ as a fraction with a common denominator, multiply by $\frac{w}{w}$ .

$\frac{w + 6}{w^{2}} \cdot \frac{6}{6} + \frac{w + 9}{6 w} \cdot \frac{w}{w}$

Step 3

Write each expression with a common denominator of $6 w^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{w + 6}{w^{2}}$ by $\frac{6}{6}$ .

$\frac{(w + 6) \cdot 6}{w^{2} \cdot 6} + \frac{w + 9}{6 w} \cdot \frac{w}{w}$

Step 3.2

Multiply $\frac{w + 9}{6 w}$ by $\frac{w}{w}$ .

$\frac{(w + 6) \cdot 6}{w^{2} \cdot 6} + \frac{(w + 9) w}{6 w \cdot w}$

Step 3.3

Raise $w$ to the power of $1$ .

$\frac{(w + 6) \cdot 6}{w^{2} \cdot 6} + \frac{(w + 9) w}{6 (w^{1} w)}$

Step 3.4

Raise $w$ to the power of $1$ .

$\frac{(w + 6) \cdot 6}{w^{2} \cdot 6} + \frac{(w + 9) w}{6 (w^{1} w^{1})}$

Step 3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(w + 6) \cdot 6}{w^{2} \cdot 6} + \frac{(w + 9) w}{6 w^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

$\frac{(w + 6) \cdot 6}{w^{2} \cdot 6} + \frac{(w + 9) w}{6 w^{2}}$

Step 3.7

Reorder the factors of $w^{2} \cdot 6$ .

$\frac{(w + 6) \cdot 6}{6 w^{2}} + \frac{(w + 9) w}{6 w^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{(w + 6) \cdot 6 + (w + 9) w}{6 w^{2}}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{w \cdot 6 + 6 \cdot 6 + (w + 9) w}{6 w^{2}}$

Step 5.2

Move $6$ to the left of $w$ .

$\frac{6 \cdot w + 6 \cdot 6 + (w + 9) w}{6 w^{2}}$

Step 5.3

Multiply $6$ by $6$ .

$\frac{6 \cdot w + 36 + (w + 9) w}{6 w^{2}}$

Step 5.4

Apply the distributive property.

$\frac{6 w + 36 + w \cdot w + 9 w}{6 w^{2}}$

Step 5.5

Multiply $w$ by $w$ .

$\frac{6 w + 36 + w^{2} + 9 w}{6 w^{2}}$

Step 5.6

Add $6 w$ and $9 w$ .

$\frac{15 w + 36 + w^{2}}{6 w^{2}}$

Step 5.7

Reorder terms.

$\frac{w^{2} + 15 w + 36}{6 w^{2}}$

Step 5.8

Factor $w^{2} + 15 w + 36$ using the AC method.

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Step 5.8.1

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 $36$ and whose sum is $15$ .

$3, 12$

Step 5.8.2

Write the factored form using these integers.

$\frac{(w + 3) (w + 12)}{6 w^{2}}$