Basic Math Examples

$\frac{16 a^{2} - 28 a + 12}{a^{4} - 3 a^{3}} \div \frac{3 - 4 a}{4 a^{3}} \cdot \frac{a - 3}{4 a - 4}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $4$ out of $16 a^{2} - 28 a + 12$ .

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Step 1.1.1

Factor $4$ out of $16 a^{2}$ .

$\frac{4 (4 a^{2}) - 28 a + 12}{a^{4} - 3 a^{3}} \div \frac{3 - 4 a}{4 a^{3}} \cdot \frac{a - 3}{4 a - 4}$

Step 1.1.2

Factor $4$ out of $- 28 a$ .

$\frac{4 (4 a^{2}) + 4 (- 7 a) + 12}{a^{4} - 3 a^{3}} \div \frac{3 - 4 a}{4 a^{3}} \cdot \frac{a - 3}{4 a - 4}$

Step 1.1.3

Factor $4$ out of $12$ .

$\frac{4 (4 a^{2}) + 4 (- 7 a) + 4 (3)}{a^{4} - 3 a^{3}} \div \frac{3 - 4 a}{4 a^{3}} \cdot \frac{a - 3}{4 a - 4}$

Step 1.1.4

Factor $4$ out of $4 (4 a^{2}) + 4 (- 7 a)$ .

$\frac{4 (4 a^{2} - 7 a) + 4 (3)}{a^{4} - 3 a^{3}} \div \frac{3 - 4 a}{4 a^{3}} \cdot \frac{a - 3}{4 a - 4}$

Step 1.1.5

Factor $4$ out of $4 (4 a^{2} - 7 a) + 4 (3)$ .

$\frac{4 (4 a^{2} - 7 a + 3)}{a^{4} - 3 a^{3}} \div \frac{3 - 4 a}{4 a^{3}} \cdot \frac{a - 3}{4 a - 4}$

Step 1.2

Factor by grouping.

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Step 1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 3 = 12$ and whose sum is $b = - 7$ .

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Step 1.2.1.1

Factor $- 7$ out of $- 7 a$ .

$\frac{4 (4 a^{2} - 7 (a) + 3)}{a^{4} - 3 a^{3}} \div \frac{3 - 4 a}{4 a^{3}} \cdot \frac{a - 3}{4 a - 4}$

Step 1.2.1.2

Rewrite $- 7$ as $- 3$ plus $- 4$

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

Step 1.2.1.3

Apply the distributive property.

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

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{4 (a (4 a - 3) - (4 a - 3))}{a^{4} - 3 a^{3}} \div \frac{3 - 4 a}{4 a^{3}} \cdot \frac{a - 3}{4 a - 4}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $4 a - 3$ .

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

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

Step 2

Simplify with factoring out.

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Step 2.1

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

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Step 2.1.1

Factor $a^{3}$ out of $a^{4}$ .

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

Step 2.1.2

Factor $a^{3}$ out of $- 3 a^{3}$ .

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

Step 2.1.3

Factor $a^{3}$ out of $a^{3} a + a^{3} \cdot - 3$ .

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

Step 2.2

Factor $4$ out of $4 a - 4$ .

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Step 2.2.1

Factor $4$ out of $4 a$ .

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

Step 2.2.2

Factor $4$ out of $- 4$ .

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

Step 2.2.3

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

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

Step 3

To divide by a fraction, multiply by its reciprocal.

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

Step 4

Cancel the common factor of $a^{3}$ .

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Step 4.1

Factor $a^{3}$ out of $4 a^{3}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Multiply $4$ by $4$ .

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

Step 7

Cancel the common factor of $4 (a - 1)$ .

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Step 7.1

Factor $4 (a - 1)$ out of $16 (4 a - 3) (a - 1)$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

$\frac{4 (4 a - 3)}{(a - 3) (3 - 4 a)} \cdot (a - 3)$

Step 8

Cancel the common factor of $a - 3$ .

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Step 8.1

Cancel the common factor.

$\frac{4 (4 a - 3)}{(a - 3) (3 - 4 a)} \cdot (a - 3)$

Step 8.2

Rewrite the expression.

$\frac{4 (4 a - 3)}{3 - 4 a}$

Step 9

Cancel the common factor of $4 a - 3$ and $3 - 4 a$ .

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Step 9.1

Factor $- 1$ out of $4 a$ .

$\frac{4 (- (- 4 a) - 3)}{3 - 4 a}$

Step 9.2

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 9.3

Factor $- 1$ out of $- (- 4 a) - 1 (3)$ .

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

Step 9.4

Rewrite $- (- 4 a + 3)$ as $- 1 (- 4 a + 3)$ .

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

Step 9.5

Reorder terms.

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

Step 9.6

Cancel the common factor.

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

Step 9.7

Divide $4 \cdot (- 1)$ by $1$ .

$4 \cdot (- 1)$

Step 10

Multiply $4$ by $- 1$ .

$- 4$