Basic Math Examples

$\frac{8 a + 1 + {(a - 1)}^{2} (2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite ${(a - 1)}^{2}$ as $(a - 1) (a - 1)$ .

$\frac{8 a + 1 + (a - 1) (a - 1) (2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.2

Expand $(a - 1) (a - 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\frac{8 a + 1 + (a (a - 1) - 1 (a - 1)) (2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.2.2

Apply the distributive property.

$\frac{8 a + 1 + (a \cdot a + a \cdot - 1 - 1 (a - 1)) (2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.2.3

Apply the distributive property.

$\frac{8 a + 1 + (a \cdot a + a \cdot - 1 - 1 a - 1 \cdot - 1) (2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $a$ by $a$ .

$\frac{8 a + 1 + (a^{2} + a \cdot - 1 - 1 a - 1 \cdot - 1) (2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.3.1.2

Move $- 1$ to the left of $a$ .

$\frac{8 a + 1 + (a^{2} - 1 \cdot a - 1 a - 1 \cdot - 1) (2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.3.1.3

Rewrite $- 1 a$ as $- a$ .

$\frac{8 a + 1 + (a^{2} - a - 1 a - 1 \cdot - 1) (2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.3.1.4

Rewrite $- 1 a$ as $- a$ .

$\frac{8 a + 1 + (a^{2} - a - a - 1 \cdot - 1) (2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

$\frac{8 a + 1 + (a^{2} - a - a + 1) (2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.3.2

Subtract $a$ from $- a$ .

$\frac{8 a + 1 + (a^{2} - 2 a + 1) (2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.4

Expand $(a^{2} - 2 a + 1) (2 a + 1)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{8 a + 1 + (a^{2} (2 a) + a^{2} \cdot 1 - 2 a (2 a) - 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5

Simplify each term.

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Step 1.5.1

Rewrite using the commutative property of multiplication.

$\frac{8 a + 1 + (2 a^{2} a + a^{2} \cdot 1 - 2 a (2 a) - 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.2

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

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Step 1.5.2.1

Move $a$ .

$\frac{8 a + 1 + (2 (a \cdot a^{2}) + a^{2} \cdot 1 - 2 a (2 a) - 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.2.2

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

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Step 1.5.2.2.1

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

$\frac{8 a + 1 + (2 (a^{1} a^{2}) + a^{2} \cdot 1 - 2 a (2 a) - 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.2.2.2

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

$\frac{8 a + 1 + (2 a^{1 + 2} + a^{2} \cdot 1 - 2 a (2 a) - 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.2.3

Add $1$ and $2$ .

$\frac{8 a + 1 + (2 a^{3} + a^{2} \cdot 1 - 2 a (2 a) - 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.3

Multiply $a^{2}$ by $1$ .

$\frac{8 a + 1 + (2 a^{3} + a^{2} - 2 a (2 a) - 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.4

Rewrite using the commutative property of multiplication.

$\frac{8 a + 1 + (2 a^{3} + a^{2} - 2 \cdot 2 a \cdot a - 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.5

Multiply $a$ by $a$ by adding the exponents.

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Step 1.5.5.1

Move $a$ .

$\frac{8 a + 1 + (2 a^{3} + a^{2} - 2 \cdot 2 (a \cdot a) - 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.5.2

Multiply $a$ by $a$ .

$\frac{8 a + 1 + (2 a^{3} + a^{2} - 2 \cdot 2 a^{2} - 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.6

Multiply $- 2$ by $2$ .

$\frac{8 a + 1 + (2 a^{3} + a^{2} - 4 a^{2} - 2 a \cdot 1 + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.7

Multiply $- 2$ by $1$ .

$\frac{8 a + 1 + (2 a^{3} + a^{2} - 4 a^{2} - 2 a + 1 (2 a) + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.8

Multiply $2 a$ by $1$ .

$\frac{8 a + 1 + (2 a^{3} + a^{2} - 4 a^{2} - 2 a + 2 a + 1 \cdot 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.5.9

Multiply $1$ by $1$ .

$\frac{8 a + 1 + (2 a^{3} + a^{2} - 4 a^{2} - 2 a + 2 a + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.6

Combine the opposite terms in $2 a^{3} + a^{2} - 4 a^{2} - 2 a + 2 a + 1$ .

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Step 1.6.1

Add $- 2 a$ and $2 a$ .

$\frac{8 a + 1 + (2 a^{3} + a^{2} - 4 a^{2} + 0 + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.6.2

Add $2 a^{3} + a^{2} - 4 a^{2}$ and $0$ .

$\frac{8 a + 1 + (2 a^{3} + a^{2} - 4 a^{2} + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.7

Subtract $4 a^{2}$ from $a^{2}$ .

$\frac{8 a + 1 + (2 a^{3} - 3 a^{2} + 1) (2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.8

Expand $(2 a^{3} - 3 a^{2} + 1) (2 a - 1)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{8 a + 1 + (2 a^{3} (2 a) + 2 a^{3} \cdot - 1 - 3 a^{2} (2 a) - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9

Simplify each term.

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Step 1.9.1

Rewrite using the commutative property of multiplication.

$\frac{8 a + 1 + (2 \cdot 2 a^{3} a + 2 a^{3} \cdot - 1 - 3 a^{2} (2 a) - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.2

Multiply $a^{3}$ by $a$ by adding the exponents.

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Step 1.9.2.1

Move $a$ .

$\frac{8 a + 1 + (2 \cdot 2 (a \cdot a^{3}) + 2 a^{3} \cdot - 1 - 3 a^{2} (2 a) - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.2.2

Multiply $a$ by $a^{3}$ .

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Step 1.9.2.2.1

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

$\frac{8 a + 1 + (2 \cdot 2 (a^{1} a^{3}) + 2 a^{3} \cdot - 1 - 3 a^{2} (2 a) - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.2.2.2

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

$\frac{8 a + 1 + (2 \cdot 2 a^{1 + 3} + 2 a^{3} \cdot - 1 - 3 a^{2} (2 a) - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.2.3

Add $1$ and $3$ .

$\frac{8 a + 1 + (2 \cdot 2 a^{4} + 2 a^{3} \cdot - 1 - 3 a^{2} (2 a) - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.3

Multiply $2$ by $2$ .

$\frac{8 a + 1 + (4 a^{4} + 2 a^{3} \cdot - 1 - 3 a^{2} (2 a) - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.4

Multiply $- 1$ by $2$ .

$\frac{8 a + 1 + (4 a^{4} - 2 a^{3} - 3 a^{2} (2 a) - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.5

Rewrite using the commutative property of multiplication.

$\frac{8 a + 1 + (4 a^{4} - 2 a^{3} - 3 \cdot 2 a^{2} a - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.6

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

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Step 1.9.6.1

Move $a$ .

$\frac{8 a + 1 + (4 a^{4} - 2 a^{3} - 3 \cdot 2 (a \cdot a^{2}) - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.6.2

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

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Step 1.9.6.2.1

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

$\frac{8 a + 1 + (4 a^{4} - 2 a^{3} - 3 \cdot 2 (a^{1} a^{2}) - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.6.2.2

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

$\frac{8 a + 1 + (4 a^{4} - 2 a^{3} - 3 \cdot 2 a^{1 + 2} - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.6.3

Add $1$ and $2$ .

$\frac{8 a + 1 + (4 a^{4} - 2 a^{3} - 3 \cdot 2 a^{3} - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.7

Multiply $- 3$ by $2$ .

$\frac{8 a + 1 + (4 a^{4} - 2 a^{3} - 6 a^{3} - 3 a^{2} \cdot - 1 + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.8

Multiply $- 1$ by $- 3$ .

$\frac{8 a + 1 + (4 a^{4} - 2 a^{3} - 6 a^{3} + 3 a^{2} + 1 (2 a) + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.9

Multiply $2 a$ by $1$ .

$\frac{8 a + 1 + (4 a^{4} - 2 a^{3} - 6 a^{3} + 3 a^{2} + 2 a + 1 \cdot - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.9.10

Multiply $- 1$ by $1$ .

$\frac{8 a + 1 + (4 a^{4} - 2 a^{3} - 6 a^{3} + 3 a^{2} + 2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.10

Subtract $6 a^{3}$ from $- 2 a^{3}$ .

$\frac{8 a + 1 + (4 a^{4} - 8 a^{3} + 3 a^{2} + 2 a - 1) \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.11

Apply the distributive property.

$\frac{8 a + 1 + 4 a^{4} \cdot 3 - 8 a^{3} \cdot 3 + 3 a^{2} \cdot 3 + 2 a \cdot 3 - 1 \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.12

Simplify.

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Step 1.12.1

Multiply $3$ by $4$ .

$\frac{8 a + 1 + 12 a^{4} - 8 a^{3} \cdot 3 + 3 a^{2} \cdot 3 + 2 a \cdot 3 - 1 \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.12.2

Multiply $3$ by $- 8$ .

$\frac{8 a + 1 + 12 a^{4} - 24 a^{3} + 3 a^{2} \cdot 3 + 2 a \cdot 3 - 1 \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.12.3

Multiply $3$ by $3$ .

$\frac{8 a + 1 + 12 a^{4} - 24 a^{3} + 9 a^{2} + 2 a \cdot 3 - 1 \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.12.4

Multiply $3$ by $2$ .

$\frac{8 a + 1 + 12 a^{4} - 24 a^{3} + 9 a^{2} + 6 a - 1 \cdot 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.12.5

Multiply $- 1$ by $3$ .

$\frac{8 a + 1 + 12 a^{4} - 24 a^{3} + 9 a^{2} + 6 a - 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.13

Add $8 a$ and $6 a$ .

$\frac{1 + 12 a^{4} - 24 a^{3} + 9 a^{2} + 14 a - 3}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 1.14

Subtract $3$ from $1$ .

$\frac{12 a^{4} - 24 a^{3} + 9 a^{2} + 14 a - 2}{(9 - 6 a) (2 a + 1) (2 a - 1)}$

Step 2

Factor $3$ out of $9 - 6 a$ .

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Step 2.1

Factor $3$ out of $9$ .

$\frac{12 a^{4} - 24 a^{3} + 9 a^{2} + 14 a - 2}{(3 (3) - 6 a) (2 a + 1) (2 a - 1)}$

Step 2.2

Factor $3$ out of $- 6 a$ .

$\frac{12 a^{4} - 24 a^{3} + 9 a^{2} + 14 a - 2}{(3 (3) + 3 (- 2 a)) (2 a + 1) (2 a - 1)}$

Step 2.3

Factor $3$ out of $3 (3) + 3 (- 2 a)$ .

$\frac{12 a^{4} - 24 a^{3} + 9 a^{2} + 14 a - 2}{3 (3 - 2 a) (2 a + 1) (2 a - 1)}$