Finite Math Examples

\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{12 a^{2} - 9 a}{27 - a^{3}} + \frac{9}{a^{2} + 3 a + 9})

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{12 a^{2} - 9 a}{27 - a^{3}} + \frac{9}{a^{2} + 3 a + 9})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Factor

$3 a$ out of

$12 a^{2} - 9 a$ .

Tap for more steps...

Step 1.1.1.1

Factor

$3 a$ out of

$12 a^{2}$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{3 a (4 a) - 9 a}{27 - a^{3}} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.1.1.2

Factor

$3 a$ out of

$- 9 a$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{3 a (4 a) + 3 a (- 3)}{27 - a^{3}} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.1.1.3

Factor

$3 a$ out of

$3 a (4 a) + 3 a (- 3)$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{3 a (4 a - 3)}{27 - a^{3}} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.1.2

Simplify the denominator.

Tap for more steps...

Step 1.1.2.1

Rewrite

$27$ as

$3^{3}$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{3 a (4 a - 3)}{3^{3} - a^{3}} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.1.2.2

Since both terms are perfect cubes, factor using the difference of cubes formula,

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where

$a = 3$ and

$b = a$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{3 a (4 a - 3)}{(3 - a) (3^{2} + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.1.2.3

Raise

$3$ to the power of

$2$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{3 a (4 a - 3)}{(3 - a) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.2

Rewrite

$3$ as

$- 1 (- 3)$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{3 a (4 a - 3)}{(- 1 (- 3) - a) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.3

Factor

$- 1$ out of

$- a$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{3 a (4 a - 3)}{(- 1 (- 3) - (a)) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.4

Factor

$- 1$ out of

$- 1 (- 3) - (a)$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{3 a (4 a - 3)}{- 1 (- 3 + a) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.5

Move a negative from the denominator of

$\frac{3 a (4 a - 3)}{- 1 (- 3 + a) (9 + 3 a + a^{2})}$ to the numerator.

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{- (3 a (4 a - 3))}{(- 3 + a) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.6

Reorder terms.

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} + \frac{- (3 a (4 a - 3))}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.7

To write

$\frac{a}{a - 3}$ as a fraction with a common denominator, multiply by

$\frac{9 + 3 a + a^{2}}{9 + 3 a + a^{2}}$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a}{a - 3} \cdot \frac{9 + 3 a + a^{2}}{9 + 3 a + a^{2}} + \frac{- (3 a (4 a - 3))}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.8

Multiply

$\frac{a}{a - 3}$ by

$\frac{9 + 3 a + a^{2}}{9 + 3 a + a^{2}}$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (9 + 3 a + a^{2})}{(a - 3) (9 + 3 a + a^{2})} + \frac{- (3 a (4 a - 3))}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.9

Combine the numerators over the common denominator.

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (9 + 3 a + a^{2}) - (3 a (4 a - 3))}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.10

Simplify each term.

Tap for more steps...

Step 1.10.1

Simplify the numerator.

Tap for more steps...

Step 1.10.1.1

Factor

$a$ out of

$a (9 + 3 a + a^{2}) - 1 \cdot 3 a (4 a - 3)$ .

Tap for more steps...

Step 1.10.1.1.1

Factor

$a$ out of

$- 1 \cdot 3 a (4 a - 3)$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (9 + 3 a + a^{2}) + a (- 1 \cdot 3 (4 a - 3))}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.10.1.1.2

Factor

$a$ out of

$a (9 + 3 a + a^{2}) + a (- 1 \cdot 3 (4 a - 3))$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (9 + 3 a + a^{2} - 1 \cdot 3 (4 a - 3))}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.10.1.2

Multiply

$- 1$ by

$3$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (9 + 3 a + a^{2} - 3 (4 a - 3))}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.10.1.3

Apply the distributive property.

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (9 + 3 a + a^{2} - 3 (4 a) - 3 \cdot - 3)}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.10.1.4

Multiply

$4$ by

$- 3$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (9 + 3 a + a^{2} - 12 a - 3 \cdot - 3)}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.10.1.5

Multiply

$- 3$ by

$- 3$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (9 + 3 a + a^{2} - 12 a + 9)}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.10.1.6

Add

$9$ and

$9$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (3 a + a^{2} - 12 a + 18)}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.10.1.7

Subtract

$12 a$ from

$3 a$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (a^{2} - 9 a + 18)}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.10.1.8

Factor

$a^{2} - 9 a + 18$ using the AC method.

Tap for more steps...

Step 1.10.1.8.1

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

$18$ and whose sum is

$- 9$ .

$- 6, - 3$

Step 1.10.1.8.2

Write the factored form using these integers.

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a ((a - 6) (a - 3))}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (a - 6) (a - 3)}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.10.2

Cancel the common factor of

$a - 3$ .

Tap for more steps...

Step 1.10.2.1

Cancel the common factor.

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (a - 6) (a - 3)}{(a - 3) (9 + 3 a + a^{2})} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.10.2.2

Rewrite the expression.

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (a - 6)}{9 + 3 a + a^{2}} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.11

Reorder terms.

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 (\frac{a (a - 6)}{a^{2} + 3 a + 9} + \frac{9}{a^{2} + 3 a + 9})$

Step 1.12

Combine the numerators over the common denominator.

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 \frac{a (a - 6) + 9}{a^{2} + 3 a + 9}$

Step 1.13

Simplify the numerator.

Tap for more steps...

Step 1.13.1

Apply the distributive property.

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 \frac{a \cdot a + a \cdot - 6 + 9}{a^{2} + 3 a + 9}$

Step 1.13.2

Multiply

$a$ by

$a$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 \frac{a^{2} + a \cdot - 6 + 9}{a^{2} + 3 a + 9}$

Step 1.13.3

Move

$- 6$ to the left of

$a$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 \frac{a^{2} - 6 \cdot a + 9}{a^{2} + 3 a + 9}$

Step 1.13.4

Factor using the perfect square rule.

Tap for more steps...

Step 1.13.4.1

Rewrite

$9$ as

$3^{2}$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 \frac{a^{2} - 6 a + 3^{2}}{a^{2} + 3 a + 9}$

Step 1.13.4.2

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

$6 a = 2 \cdot a \cdot 3$

Step 1.13.4.3

Rewrite the polynomial.

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 \frac{a^{2} - 2 \cdot a \cdot 3 + 3^{2}}{a^{2} + 3 a + 9}$

Step 1.13.4.4

Factor using the perfect square trinomial rule

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

$a = a$ and

$b = 3$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - 1 \frac{{(a - 3)}^{2}}{a^{2} + 3 a + 9}$

Step 1.14

Rewrite

$- 1 \frac{{(a - 3)}^{2}}{a^{2} + 3 a + 9}$ as

$- \frac{{(a - 3)}^{2}}{a^{2} + 3 a + 9}$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} - \frac{{(a - 3)}^{2}}{a^{2} + 3 a + 9}$

Step 2

To write

$\frac{9 a^{2}}{{(3 - a)}^{2}}$ as a fraction with a common denominator, multiply by

$\frac{a^{2} + 3 a + 9}{a^{2} + 3 a + 9}$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} \cdot \frac{a^{2} + 3 a + 9}{a^{2} + 3 a + 9} - \frac{{(a - 3)}^{2}}{a^{2} + 3 a + 9}$

Step 3

To write

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

$\frac{{(3 - a)}^{2}}{{(3 - a)}^{2}}$ .

$\frac{9 a^{2}}{{(3 - a)}^{2}} \cdot \frac{a^{2} + 3 a + 9}{a^{2} + 3 a + 9} - \frac{{(a - 3)}^{2}}{a^{2} + 3 a + 9} \cdot \frac{{(3 - a)}^{2}}{{(3 - a)}^{2}}$

Step 4

Write each expression with a common denominator of

$(a^{2} + 3 a + 9) {(3 - a)}^{2}$ , by multiplying each by an appropriate factor of

$1$ .

Tap for more steps...

Step 4.1

Multiply

$\frac{9 a^{2}}{{(3 - a)}^{2}}$ by

$\frac{a^{2} + 3 a + 9}{a^{2} + 3 a + 9}$ .

$\frac{9 a^{2} (a^{2} + 3 a + 9)}{{(3 - a)}^{2} (a^{2} + 3 a + 9)} - \frac{{(a - 3)}^{2}}{a^{2} + 3 a + 9} \cdot \frac{{(3 - a)}^{2}}{{(3 - a)}^{2}}$

Step 4.2

Multiply

$\frac{{(a - 3)}^{2}}{a^{2} + 3 a + 9}$ by

$\frac{{(3 - a)}^{2}}{{(3 - a)}^{2}}$ .

$\frac{9 a^{2} (a^{2} + 3 a + 9)}{{(3 - a)}^{2} (a^{2} + 3 a + 9)} - \frac{{(a - 3)}^{2} {(3 - a)}^{2}}{(a^{2} + 3 a + 9) {(3 - a)}^{2}}$

Step 4.3

Reorder the factors of

$(a^{2} + 3 a + 9) {(3 - a)}^{2}$ .

$\frac{9 a^{2} (a^{2} + 3 a + 9)}{{(3 - a)}^{2} (a^{2} + 3 a + 9)} - \frac{{(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 5

Combine the numerators over the common denominator.

$\frac{9 a^{2} (a^{2} + 3 a + 9) - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Apply the distributive property.

$\frac{9 a^{2} a^{2} + 9 a^{2} (3 a) + 9 a^{2} \cdot 9 - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

Multiply

$a^{2}$ by

$a^{2}$ by adding the exponents.

Tap for more steps...

Step 6.2.1.1

Move

$a^{2}$ .

$\frac{9 (a^{2} a^{2}) + 9 a^{2} (3 a) + 9 a^{2} \cdot 9 - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.2.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 a^{2 + 2} + 9 a^{2} (3 a) + 9 a^{2} \cdot 9 - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.2.1.3

Add

$2$ and

$2$ .

$\frac{9 a^{4} + 9 a^{2} (3 a) + 9 a^{2} \cdot 9 - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.2.2

Rewrite using the commutative property of multiplication.

$\frac{9 a^{4} + 9 \cdot 3 a^{2} a + 9 a^{2} \cdot 9 - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.2.3

Multiply

$9$ by

$9$ .

$\frac{9 a^{4} + 9 \cdot 3 a^{2} a + 81 a^{2} - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.3

Simplify each term.

Tap for more steps...

Step 6.3.1

Multiply

$a^{2}$ by

$a$ by adding the exponents.

Tap for more steps...

Step 6.3.1.1

Move

$a$ .

$\frac{9 a^{4} + 9 \cdot 3 (a \cdot a^{2}) + 81 a^{2} - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.3.1.2

Multiply

$a$ by

$a^{2}$ .

Tap for more steps...

Step 6.3.1.2.1

Raise

$a$ to the power of

$1$ .

$\frac{9 a^{4} + 9 \cdot 3 (a^{1} a^{2}) + 81 a^{2} - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.3.1.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 a^{4} + 9 \cdot 3 a^{1 + 2} + 81 a^{2} - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.3.1.3

Add

$1$ and

$2$ .

$\frac{9 a^{4} + 9 \cdot 3 a^{3} + 81 a^{2} - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.3.2

Multiply

$9$ by

$3$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - {(a - 3)}^{2} {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.4

Rewrite

${(a - 3)}^{2}$ as

$(a - 3) (a - 3)$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - ((a - 3) (a - 3)) {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.5

Expand

$(a - 3) (a - 3)$ using the FOIL Method.

Tap for more steps...

Step 6.5.1

Apply the distributive property.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - (a (a - 3) - 3 (a - 3)) {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.5.2

Apply the distributive property.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - (a \cdot a + a \cdot - 3 - 3 (a - 3)) {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.5.3

Apply the distributive property.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - (a \cdot a + a \cdot - 3 - 3 a - 3 \cdot - 3) {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.6

Simplify and combine like terms.

Tap for more steps...

Step 6.6.1

Simplify each term.

Tap for more steps...

Step 6.6.1.1

Multiply

$a$ by

$a$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - (a^{2} + a \cdot - 3 - 3 a - 3 \cdot - 3) {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.6.1.2

Move

$- 3$ to the left of

$a$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - (a^{2} - 3 \cdot a - 3 a - 3 \cdot - 3) {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.6.1.3

Multiply

$- 3$ by

$- 3$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - (a^{2} - 3 a - 3 a + 9) {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.6.2

Subtract

$3 a$ from

$- 3 a$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - (a^{2} - 6 a + 9) {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.7

Apply the distributive property.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} - (- 6 a) - 1 \cdot 9) {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.8

Simplify.

Tap for more steps...

Step 6.8.1

Multiply

$- 6$ by

$- 1$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 1 \cdot 9) {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.8.2

Multiply

$- 1$ by

$9$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) {(3 - a)}^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.9

Rewrite

${(3 - a)}^{2}$ as

$(3 - a) (3 - a)$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) ((3 - a) (3 - a))}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.10

Expand

$(3 - a) (3 - a)$ using the FOIL Method.

Tap for more steps...

Step 6.10.1

Apply the distributive property.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (3 (3 - a) - a (3 - a))}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.10.2

Apply the distributive property.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (3 \cdot 3 + 3 (- a) - a (3 - a))}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.10.3

Apply the distributive property.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (3 \cdot 3 + 3 (- a) - a \cdot 3 - a (- a))}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.11

Simplify and combine like terms.

Tap for more steps...

Step 6.11.1

Simplify each term.

Tap for more steps...

Step 6.11.1.1

Multiply

$3$ by

$3$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (9 + 3 (- a) - a \cdot 3 - a (- a))}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.11.1.2

Multiply

$- 1$ by

$3$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (9 - 3 a - a \cdot 3 - a (- a))}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.11.1.3

Multiply

$3$ by

$- 1$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (9 - 3 a - 3 a - a (- a))}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.11.1.4

Rewrite using the commutative property of multiplication.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (9 - 3 a - 3 a - 1 \cdot - 1 a \cdot a)}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.11.1.5

Multiply

$a$ by

$a$ by adding the exponents.

Tap for more steps...

Step 6.11.1.5.1

Move

$a$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (9 - 3 a - 3 a - 1 \cdot - 1 (a \cdot a))}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.11.1.5.2

Multiply

$a$ by

$a$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (9 - 3 a - 3 a - 1 \cdot - 1 a^{2})}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.11.1.6

Multiply

$- 1$ by

$- 1$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (9 - 3 a - 3 a + 1 a^{2})}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.11.1.7

Multiply

$a^{2}$ by

$1$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (9 - 3 a - 3 a + a^{2})}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.11.2

Subtract

$3 a$ from

$- 3 a$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + (- a^{2} + 6 a - 9) (9 - 6 a + a^{2})}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.12

Expand

$(- a^{2} + 6 a - 9) (9 - 6 a + a^{2})$ by multiplying each term in the first expression by each term in the second expression.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - a^{2} \cdot 9 - a^{2} (- 6 a) - a^{2} a^{2} + 6 a \cdot 9 + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13

Simplify each term.

Tap for more steps...

Step 6.13.1

Multiply

$9$ by

$- 1$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} - a^{2} (- 6 a) - a^{2} a^{2} + 6 a \cdot 9 + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.2

Rewrite using the commutative property of multiplication.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} - 1 \cdot - 6 a^{2} a - a^{2} a^{2} + 6 a \cdot 9 + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.3

Multiply

$a^{2}$ by

$a$ by adding the exponents.

Tap for more steps...

Step 6.13.3.1

Move

$a$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} - 1 \cdot - 6 (a \cdot a^{2}) - a^{2} a^{2} + 6 a \cdot 9 + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.3.2

Multiply

$a$ by

$a^{2}$ .

Tap for more steps...

Step 6.13.3.2.1

Raise

$a$ to the power of

$1$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} - 1 \cdot - 6 (a^{1} a^{2}) - a^{2} a^{2} + 6 a \cdot 9 + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.3.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} - 1 \cdot - 6 a^{1 + 2} - a^{2} a^{2} + 6 a \cdot 9 + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.3.3

Add

$1$ and

$2$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} - 1 \cdot - 6 a^{3} - a^{2} a^{2} + 6 a \cdot 9 + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.4

Multiply

$- 1$ by

$- 6$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{2} a^{2} + 6 a \cdot 9 + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.5

Multiply

$a^{2}$ by

$a^{2}$ by adding the exponents.

Tap for more steps...

Step 6.13.5.1

Move

$a^{2}$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - (a^{2} a^{2}) + 6 a \cdot 9 + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.5.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{2 + 2} + 6 a \cdot 9 + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.5.3

Add

$2$ and

$2$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 6 a \cdot 9 + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.6

Multiply

$9$ by

$6$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 54 a + 6 a (- 6 a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.7

Rewrite using the commutative property of multiplication.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 54 a + 6 \cdot - 6 a \cdot a + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.8

Multiply

$a$ by

$a$ by adding the exponents.

Tap for more steps...

Step 6.13.8.1

Move

$a$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 54 a + 6 \cdot - 6 (a \cdot a) + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.8.2

Multiply

$a$ by

$a$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 54 a + 6 \cdot - 6 a^{2} + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.9

Multiply

$6$ by

$- 6$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 54 a - 36 a^{2} + 6 a \cdot a^{2} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.10

Multiply

$a$ by

$a^{2}$ by adding the exponents.

Tap for more steps...

Step 6.13.10.1

Move

$a^{2}$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 54 a - 36 a^{2} + 6 (a^{2} a) - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.10.2

Multiply

$a^{2}$ by

$a$ .

Tap for more steps...

Step 6.13.10.2.1

Raise

$a$ to the power of

$1$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 54 a - 36 a^{2} + 6 (a^{2} a^{1}) - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.10.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 54 a - 36 a^{2} + 6 a^{2 + 1} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.10.3

Add

$2$ and

$1$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 54 a - 36 a^{2} + 6 a^{3} - 9 \cdot 9 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.11

Multiply

$- 9$ by

$9$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 54 a - 36 a^{2} + 6 a^{3} - 81 - 9 (- 6 a) - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.13.12

Multiply

$- 6$ by

$- 9$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} - 9 a^{2} + 6 a^{3} - a^{4} + 54 a - 36 a^{2} + 6 a^{3} - 81 + 54 a - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.14

Subtract

$36 a^{2}$ from

$- 9 a^{2}$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + 6 a^{3} - a^{4} + 54 a - 45 a^{2} + 6 a^{3} - 81 + 54 a - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.15

Add

$6 a^{3}$ and

$6 a^{3}$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + 12 a^{3} - a^{4} + 54 a - 45 a^{2} - 81 + 54 a - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.16

Add

$54 a$ and

$54 a$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + 12 a^{3} - a^{4} + 108 a - 45 a^{2} - 81 - 9 a^{2}}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.17

Subtract

$9 a^{2}$ from

$- 45 a^{2}$ .

$\frac{9 a^{4} + 27 a^{3} + 81 a^{2} + 12 a^{3} - a^{4} + 108 a - 54 a^{2} - 81}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.18

Subtract

$a^{4}$ from

$9 a^{4}$ .

$\frac{8 a^{4} + 27 a^{3} + 81 a^{2} + 12 a^{3} + 108 a - 54 a^{2} - 81}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.19

Add

$27 a^{3}$ and

$12 a^{3}$ .

$\frac{8 a^{4} + 39 a^{3} + 81 a^{2} + 108 a - 54 a^{2} - 81}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$

Step 6.20

Subtract

$54 a^{2}$ from

$81 a^{2}$ .

$\frac{8 a^{4} + 39 a^{3} + 27 a^{2} + 108 a - 81}{{(3 - a)}^{2} (a^{2} + 3 a + 9)}$