Basic Math Examples

$\frac{w}{w^{2} + 6 w + 9} + \frac{3}{w^{2} + 9 w} + 18$

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $9$ as $3^{2}$ .

$\frac{w}{w^{2} + 6 w + 3^{2}} + \frac{3}{w^{2} + 9 w} + 18$

Step 1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 w = 2 \cdot w \cdot 3$

Step 1.1.3

Rewrite the polynomial.

$\frac{w}{w^{2} + 2 \cdot w \cdot 3 + 3^{2}} + \frac{3}{w^{2} + 9 w} + 18$

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = w$ and $b = 3$ .

$\frac{w}{{(w + 3)}^{2}} + \frac{3}{w^{2} + 9 w} + 18$

Step 1.2

Factor $w$ out of $w^{2} + 9 w$ .

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Step 1.2.1

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

$\frac{w}{{(w + 3)}^{2}} + \frac{3}{w \cdot w + 9 w} + 18$

Step 1.2.2

Factor $w$ out of $9 w$ .

$\frac{w}{{(w + 3)}^{2}} + \frac{3}{w \cdot w + w \cdot 9} + 18$

Step 1.2.3

Factor $w$ out of $w \cdot w + w \cdot 9$ .

$\frac{w}{{(w + 3)}^{2}} + \frac{3}{w (w + 9)} + 18$

Step 2

To write $\frac{w}{{(w + 3)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{w (w + 9)}{w (w + 9)}$ .

$\frac{w}{{(w + 3)}^{2}} \cdot \frac{w (w + 9)}{w (w + 9)} + \frac{3}{w (w + 9)} + 18$

Step 3

To write $\frac{3}{w (w + 9)}$ as a fraction with a common denominator, multiply by $\frac{{(w + 3)}^{2}}{{(w + 3)}^{2}}$ .

$\frac{w}{{(w + 3)}^{2}} \cdot \frac{w (w + 9)}{w (w + 9)} + \frac{3}{w (w + 9)} \cdot \frac{{(w + 3)}^{2}}{{(w + 3)}^{2}} + 18$

Step 4

Write each expression with a common denominator of $w (w + 9) {(w + 3)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{w}{{(w + 3)}^{2}}$ by $\frac{w (w + 9)}{w (w + 9)}$ .

$\frac{w (w (w + 9))}{{(w + 3)}^{2} (w (w + 9))} + \frac{3}{w (w + 9)} \cdot \frac{{(w + 3)}^{2}}{{(w + 3)}^{2}} + 18$

Step 4.2

Multiply $\frac{3}{w (w + 9)}$ by $\frac{{(w + 3)}^{2}}{{(w + 3)}^{2}}$ .

$\frac{w (w (w + 9))}{{(w + 3)}^{2} (w (w + 9))} + \frac{3 {(w + 3)}^{2}}{w (w + 9) {(w + 3)}^{2}} + 18$

Step 4.3

Reorder the factors of ${(w + 3)}^{2} (w (w + 9))$ .

$\frac{w (w (w + 9))}{w {(w + 3)}^{2} (w + 9)} + \frac{3 {(w + 3)}^{2}}{w (w + 9) {(w + 3)}^{2}} + 18$

Step 4.4

Reorder the factors of $w (w + 9) {(w + 3)}^{2}$ .

$\frac{w (w (w + 9))}{w {(w + 3)}^{2} (w + 9)} + \frac{3 {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 5

Combine the numerators over the common denominator.

$\frac{w (w (w + 9)) + 3 {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $w$ by $w$ by adding the exponents.

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Step 6.1.1

Move $w$ .

$\frac{w \cdot w (w + 9) + 3 {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.1.2

Multiply $w$ by $w$ .

$\frac{w^{2} (w + 9) + 3 {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.2

Apply the distributive property.

$\frac{w^{2} w + w^{2} \cdot 9 + 3 {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.3

Multiply $w^{2}$ by $w$ by adding the exponents.

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Step 6.3.1

Multiply $w^{2}$ by $w$ .

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Step 6.3.1.1

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

$\frac{w^{2} w^{1} + w^{2} \cdot 9 + 3 {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.3.1.2

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

$\frac{w^{2 + 1} + w^{2} \cdot 9 + 3 {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.3.2

Add $2$ and $1$ .

$\frac{w^{3} + w^{2} \cdot 9 + 3 {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.4

Move $9$ to the left of $w^{2}$ .

$\frac{w^{3} + 9 \cdot w^{2} + 3 {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.5

Rewrite ${(w + 3)}^{2}$ as $(w + 3) (w + 3)$ .

$\frac{w^{3} + 9 w^{2} + 3 ((w + 3) (w + 3))}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.6

Expand $(w + 3) (w + 3)$ using the FOIL Method.

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Step 6.6.1

Apply the distributive property.

$\frac{w^{3} + 9 w^{2} + 3 (w (w + 3) + 3 (w + 3))}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.6.2

Apply the distributive property.

$\frac{w^{3} + 9 w^{2} + 3 (w \cdot w + w \cdot 3 + 3 (w + 3))}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.6.3

Apply the distributive property.

$\frac{w^{3} + 9 w^{2} + 3 (w \cdot w + w \cdot 3 + 3 w + 3 \cdot 3)}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.7

Simplify and combine like terms.

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Step 6.7.1

Simplify each term.

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Step 6.7.1.1

Multiply $w$ by $w$ .

$\frac{w^{3} + 9 w^{2} + 3 (w^{2} + w \cdot 3 + 3 w + 3 \cdot 3)}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.7.1.2

Move $3$ to the left of $w$ .

$\frac{w^{3} + 9 w^{2} + 3 (w^{2} + 3 \cdot w + 3 w + 3 \cdot 3)}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.7.1.3

Multiply $3$ by $3$ .

$\frac{w^{3} + 9 w^{2} + 3 (w^{2} + 3 w + 3 w + 9)}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.7.2

Add $3 w$ and $3 w$ .

$\frac{w^{3} + 9 w^{2} + 3 (w^{2} + 6 w + 9)}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.8

Apply the distributive property.

$\frac{w^{3} + 9 w^{2} + 3 w^{2} + 3 (6 w) + 3 \cdot 9}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.9

Simplify.

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Step 6.9.1

Multiply $6$ by $3$ .

$\frac{w^{3} + 9 w^{2} + 3 w^{2} + 18 w + 3 \cdot 9}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.9.2

Multiply $3$ by $9$ .

$\frac{w^{3} + 9 w^{2} + 3 w^{2} + 18 w + 27}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 6.10

Add $9 w^{2}$ and $3 w^{2}$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27}{w {(w + 3)}^{2} (w + 9)} + 18$

Step 7

To write $18$ as a fraction with a common denominator, multiply by $\frac{w (w + 9) {(w + 3)}^{2}}{w (w + 9) {(w + 3)}^{2}}$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27}{w {(w + 3)}^{2} (w + 9)} + 18 \cdot \frac{w (w + 9) {(w + 3)}^{2}}{w (w + 9) {(w + 3)}^{2}}$

Step 8

Write each expression with a common denominator of $w (w + 9) {(w + 3)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Combine $18$ and $\frac{w (w + 9) {(w + 3)}^{2}}{w (w + 9) {(w + 3)}^{2}}$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27}{w {(w + 3)}^{2} (w + 9)} + \frac{18 (w (w + 9) {(w + 3)}^{2})}{w (w + 9) {(w + 3)}^{2}}$

Step 8.2

Reorder the factors of $w (w + 9) {(w + 3)}^{2}$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27}{w {(w + 3)}^{2} (w + 9)} + \frac{18 (w (w + 9) {(w + 3)}^{2})}{w {(w + 3)}^{2} (w + 9)}$

Step 9

Combine the numerators over the common denominator.

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 (w (w + 9) {(w + 3)}^{2})}{w {(w + 3)}^{2} (w + 9)}$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 w \cdot w + 18 w \cdot 9) {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)}$

Step 10.2

Multiply $w$ by $w$ by adding the exponents.

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Step 10.2.1

Move $w$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 (w \cdot w) + 18 w \cdot 9) {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)}$

Step 10.2.2

Multiply $w$ by $w$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 w^{2} + 18 w \cdot 9) {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)}$

Step 10.3

Multiply $9$ by $18$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 w^{2} + 162 w) {(w + 3)}^{2}}{w {(w + 3)}^{2} (w + 9)}$

Step 10.4

Rewrite ${(w + 3)}^{2}$ as $(w + 3) (w + 3)$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 w^{2} + 162 w) ((w + 3) (w + 3))}{w {(w + 3)}^{2} (w + 9)}$

Step 10.5

Expand $(w + 3) (w + 3)$ using the FOIL Method.

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Step 10.5.1

Apply the distributive property.

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 w^{2} + 162 w) (w (w + 3) + 3 (w + 3))}{w {(w + 3)}^{2} (w + 9)}$

Step 10.5.2

Apply the distributive property.

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 w^{2} + 162 w) (w \cdot w + w \cdot 3 + 3 (w + 3))}{w {(w + 3)}^{2} (w + 9)}$

Step 10.5.3

Apply the distributive property.

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 w^{2} + 162 w) (w \cdot w + w \cdot 3 + 3 w + 3 \cdot 3)}{w {(w + 3)}^{2} (w + 9)}$

Step 10.6

Simplify and combine like terms.

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Step 10.6.1

Simplify each term.

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Step 10.6.1.1

Multiply $w$ by $w$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 w^{2} + 162 w) (w^{2} + w \cdot 3 + 3 w + 3 \cdot 3)}{w {(w + 3)}^{2} (w + 9)}$

Step 10.6.1.2

Move $3$ to the left of $w$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 w^{2} + 162 w) (w^{2} + 3 \cdot w + 3 w + 3 \cdot 3)}{w {(w + 3)}^{2} (w + 9)}$

Step 10.6.1.3

Multiply $3$ by $3$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 w^{2} + 162 w) (w^{2} + 3 w + 3 w + 9)}{w {(w + 3)}^{2} (w + 9)}$

Step 10.6.2

Add $3 w$ and $3 w$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + (18 w^{2} + 162 w) (w^{2} + 6 w + 9)}{w {(w + 3)}^{2} (w + 9)}$

Step 10.7

Expand $(18 w^{2} + 162 w) (w^{2} + 6 w + 9)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{2} w^{2} + 18 w^{2} (6 w) + 18 w^{2} \cdot 9 + 162 w \cdot w^{2} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8

Simplify each term.

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Step 10.8.1

Multiply $w^{2}$ by $w^{2}$ by adding the exponents.

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Step 10.8.1.1

Move $w^{2}$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 (w^{2} w^{2}) + 18 w^{2} (6 w) + 18 w^{2} \cdot 9 + 162 w \cdot w^{2} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.1.2

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

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{2 + 2} + 18 w^{2} (6 w) + 18 w^{2} \cdot 9 + 162 w \cdot w^{2} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.1.3

Add $2$ and $2$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 18 w^{2} (6 w) + 18 w^{2} \cdot 9 + 162 w \cdot w^{2} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.2

Rewrite using the commutative property of multiplication.

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 18 \cdot 6 w^{2} w + 18 w^{2} \cdot 9 + 162 w \cdot w^{2} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.3

Multiply $w^{2}$ by $w$ by adding the exponents.

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Step 10.8.3.1

Move $w$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 18 \cdot 6 (w \cdot w^{2}) + 18 w^{2} \cdot 9 + 162 w \cdot w^{2} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.3.2

Multiply $w$ by $w^{2}$ .

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Step 10.8.3.2.1

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

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 18 \cdot 6 (w^{1} w^{2}) + 18 w^{2} \cdot 9 + 162 w \cdot w^{2} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.3.2.2

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

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 18 \cdot 6 w^{1 + 2} + 18 w^{2} \cdot 9 + 162 w \cdot w^{2} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.3.3

Add $1$ and $2$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 18 \cdot 6 w^{3} + 18 w^{2} \cdot 9 + 162 w \cdot w^{2} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.4

Multiply $18$ by $6$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 108 w^{3} + 18 w^{2} \cdot 9 + 162 w \cdot w^{2} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.5

Multiply $9$ by $18$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 108 w^{3} + 162 w^{2} + 162 w \cdot w^{2} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.6

Multiply $w$ by $w^{2}$ by adding the exponents.

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Step 10.8.6.1

Move $w^{2}$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 108 w^{3} + 162 w^{2} + 162 (w^{2} w) + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.6.2

Multiply $w^{2}$ by $w$ .

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Step 10.8.6.2.1

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

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 108 w^{3} + 162 w^{2} + 162 (w^{2} w^{1}) + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.6.2.2

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

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 108 w^{3} + 162 w^{2} + 162 w^{2 + 1} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.6.3

Add $2$ and $1$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 108 w^{3} + 162 w^{2} + 162 w^{3} + 162 w (6 w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.7

Rewrite using the commutative property of multiplication.

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 108 w^{3} + 162 w^{2} + 162 w^{3} + 162 \cdot 6 w \cdot w + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.8

Multiply $w$ by $w$ by adding the exponents.

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Step 10.8.8.1

Move $w$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 108 w^{3} + 162 w^{2} + 162 w^{3} + 162 \cdot 6 (w \cdot w) + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.8.2

Multiply $w$ by $w$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 108 w^{3} + 162 w^{2} + 162 w^{3} + 162 \cdot 6 w^{2} + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.9

Multiply $162$ by $6$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 108 w^{3} + 162 w^{2} + 162 w^{3} + 972 w^{2} + 162 w \cdot 9}{w {(w + 3)}^{2} (w + 9)}$

Step 10.8.10

Multiply $9$ by $162$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 108 w^{3} + 162 w^{2} + 162 w^{3} + 972 w^{2} + 1458 w}{w {(w + 3)}^{2} (w + 9)}$

Step 10.9

Add $108 w^{3}$ and $162 w^{3}$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 270 w^{3} + 162 w^{2} + 972 w^{2} + 1458 w}{w {(w + 3)}^{2} (w + 9)}$

Step 10.10

Add $162 w^{2}$ and $972 w^{2}$ .

$\frac{w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 270 w^{3} + 1134 w^{2} + 1458 w}{w {(w + 3)}^{2} (w + 9)}$

Step 10.11

Add $w^{3}$ and $270 w^{3}$ .

$\frac{271 w^{3} + 12 w^{2} + 18 w + 27 + 18 w^{4} + 1134 w^{2} + 1458 w}{w {(w + 3)}^{2} (w + 9)}$

Step 10.12

Add $12 w^{2}$ and $1134 w^{2}$ .

$\frac{271 w^{3} + 1146 w^{2} + 18 w + 27 + 18 w^{4} + 1458 w}{w {(w + 3)}^{2} (w + 9)}$

Step 10.13

Add $18 w$ and $1458 w$ .

$\frac{271 w^{3} + 1146 w^{2} + 1476 w + 27 + 18 w^{4}}{w {(w + 3)}^{2} (w + 9)}$

Step 10.14

Reorder terms.

$\frac{18 w^{4} + 271 w^{3} + 1146 w^{2} + 1476 w + 27}{w {(w + 3)}^{2} (w + 9)}$