Basic Math Examples

$\frac{8 a}{a^{2} - 3 a - 4} - \frac{32}{a^{2} \cdot (3 a) - 4}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Factor $a^{2} - 3 a - 4$ using the AC method.

Tap for more steps...

Step 1.1.1

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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 1.1.2

Write the factored form using these integers.

$\frac{8 a}{(a - 4) (a + 1)} - \frac{32}{a^{2} \cdot (3 a) - 4}$

Step 1.2

Simplify the denominator.

Tap for more steps...

Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{8 a}{(a - 4) (a + 1)} - \frac{32}{3 a^{2} a - 4}$

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

Move $a$ .

$\frac{8 a}{(a - 4) (a + 1)} - \frac{32}{3 (a \cdot a^{2}) - 4}$

Step 1.2.2.2

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

Tap for more steps...

Step 1.2.2.2.1

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

$\frac{8 a}{(a - 4) (a + 1)} - \frac{32}{3 (a^{1} a^{2}) - 4}$

Step 1.2.2.2.2

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

$\frac{8 a}{(a - 4) (a + 1)} - \frac{32}{3 a^{1 + 2} - 4}$

Step 1.2.2.3

Add $1$ and $2$ .

$\frac{8 a}{(a - 4) (a + 1)} - \frac{32}{3 a^{3} - 4}$

Step 2

To write $\frac{8 a}{(a - 4) (a + 1)}$ as a fraction with a common denominator, multiply by $\frac{3 a^{3} - 4}{3 a^{3} - 4}$ .

$\frac{8 a}{(a - 4) (a + 1)} \cdot \frac{3 a^{3} - 4}{3 a^{3} - 4} - \frac{32}{3 a^{3} - 4}$

Step 3

To write $- \frac{32}{3 a^{3} - 4}$ as a fraction with a common denominator, multiply by $\frac{(a - 4) (a + 1)}{(a - 4) (a + 1)}$ .

$\frac{8 a}{(a - 4) (a + 1)} \cdot \frac{3 a^{3} - 4}{3 a^{3} - 4} - \frac{32}{3 a^{3} - 4} \cdot \frac{(a - 4) (a + 1)}{(a - 4) (a + 1)}$

Step 4

Write each expression with a common denominator of $(a - 4) (a + 1) (3 a^{3} - 4)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1

Multiply $\frac{8 a}{(a - 4) (a + 1)}$ by $\frac{3 a^{3} - 4}{3 a^{3} - 4}$ .

$\frac{8 a (3 a^{3} - 4)}{(a - 4) (a + 1) (3 a^{3} - 4)} - \frac{32}{3 a^{3} - 4} \cdot \frac{(a - 4) (a + 1)}{(a - 4) (a + 1)}$

Step 4.2

Multiply $\frac{32}{3 a^{3} - 4}$ by $\frac{(a - 4) (a + 1)}{(a - 4) (a + 1)}$ .

$\frac{8 a (3 a^{3} - 4)}{(a - 4) (a + 1) (3 a^{3} - 4)} - \frac{32 ((a - 4) (a + 1))}{(3 a^{3} - 4) ((a - 4) (a + 1))}$

Step 4.3

Reorder the factors of $(a - 4) (a + 1) (3 a^{3} - 4)$ .

$\frac{8 a (3 a^{3} - 4)}{(3 a^{3} - 4) (a + 1) (a - 4)} - \frac{32 ((a - 4) (a + 1))}{(3 a^{3} - 4) ((a - 4) (a + 1))}$

Step 4.4

Reorder the factors of $(3 a^{3} - 4) ((a - 4) (a + 1))$ .

$\frac{8 a (3 a^{3} - 4)}{(3 a^{3} - 4) (a + 1) (a - 4)} - \frac{32 ((a - 4) (a + 1))}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{8 a (3 a^{3} - 4) - 32 ((a - 4) (a + 1))}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Factor $8$ out of $8 a (3 a^{3} - 4) - 32 (a - 4) (a + 1)$ .

Tap for more steps...

Step 6.1.1

Factor $8$ out of $8 a (3 a^{3} - 4)$ .

$\frac{8 (a (3 a^{3} - 4)) - 32 (a - 4) (a + 1)}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.1.2

Factor $8$ out of $- 32 (a - 4) (a + 1)$ .

$\frac{8 (a (3 a^{3} - 4)) + 8 (- 4 (a - 4) (a + 1))}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.1.3

Factor $8$ out of $8 (a (3 a^{3} - 4)) + 8 (- 4 (a - 4) (a + 1))$ .

$\frac{8 (a (3 a^{3} - 4) - 4 (a - 4) (a + 1))}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.2

Apply the distributive property.

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

Step 6.3

Rewrite using the commutative property of multiplication.

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

Step 6.4

Move $- 4$ to the left of $a$ .

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

Step 6.5

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

Tap for more steps...

Step 6.5.1

Move $a^{3}$ .

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

Step 6.5.2

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

Tap for more steps...

Step 6.5.2.1

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

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

Step 6.5.2.2

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

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

Step 6.5.3

Add $3$ and $1$ .

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

$\frac{8 (3 a^{4} - 4 a - 4 (a - 4) (a + 1))}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.6

Apply the distributive property.

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

Step 6.7

Multiply $- 4$ by $- 4$ .

$\frac{8 (3 a^{4} - 4 a + (- 4 a + 16) (a + 1))}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.8

Expand $(- 4 a + 16) (a + 1)$ using the FOIL Method.

Tap for more steps...

Step 6.8.1

Apply the distributive property.

$\frac{8 (3 a^{4} - 4 a - 4 a (a + 1) + 16 (a + 1))}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.8.2

Apply the distributive property.

$\frac{8 (3 a^{4} - 4 a - 4 a \cdot a - 4 a \cdot 1 + 16 (a + 1))}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.8.3

Apply the distributive property.

$\frac{8 (3 a^{4} - 4 a - 4 a \cdot a - 4 a \cdot 1 + 16 a + 16 \cdot 1)}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.9

Simplify and combine like terms.

Tap for more steps...

Step 6.9.1

Simplify each term.

Tap for more steps...

Step 6.9.1.1

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

Tap for more steps...

Step 6.9.1.1.1

Move $a$ .

$\frac{8 (3 a^{4} - 4 a - 4 (a \cdot a) - 4 a \cdot 1 + 16 a + 16 \cdot 1)}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.9.1.1.2

Multiply $a$ by $a$ .

$\frac{8 (3 a^{4} - 4 a - 4 a^{2} - 4 a \cdot 1 + 16 a + 16 \cdot 1)}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.9.1.2

Multiply $- 4$ by $1$ .

$\frac{8 (3 a^{4} - 4 a - 4 a^{2} - 4 a + 16 a + 16 \cdot 1)}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.9.1.3

Multiply $16$ by $1$ .

$\frac{8 (3 a^{4} - 4 a - 4 a^{2} - 4 a + 16 a + 16)}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.9.2

Add $- 4 a$ and $16 a$ .

$\frac{8 (3 a^{4} - 4 a - 4 a^{2} + 12 a + 16)}{(3 a^{3} - 4) (a + 1) (a - 4)}$

Step 6.10

Add $- 4 a$ and $12 a$ .

$\frac{8 (3 a^{4} - 4 a^{2} + 8 a + 16)}{(3 a^{3} - 4) (a + 1) (a - 4)}$