Basic Math Examples

$\frac{8 r^{3} + s^{3}}{2 r^{2} \cdot (3 r s) + s^{2}} \div \frac{12 r^{2} - 6 r + 3 s^{2}}{4 r^{2} + 4 r s + s^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{8 r^{3} + s^{3}}{2 r^{2} \cdot (3 r s) + s^{2}} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Rewrite $8 r^{3}$ as ${(2 r)}^{3}$ .

$\frac{{(2 r)}^{3} + s^{3}}{2 r^{2} \cdot (3 r s) + s^{2}} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 2.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 2 r$ and $b = s$ .

$\frac{(2 r + s) ({(2 r)}^{2} - (2 r) s + s^{2})}{2 r^{2} \cdot (3 r s) + s^{2}} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Apply the product rule to $2 r$ .

$\frac{(2 r + s) (2^{2} r^{2} - (2 r) s + s^{2})}{2 r^{2} \cdot (3 r s) + s^{2}} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 2.3.2

Raise $2$ to the power of $2$ .

$\frac{(2 r + s) (4 r^{2} - (2 r) s + s^{2})}{2 r^{2} \cdot (3 r s) + s^{2}} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 2.3.3

Multiply $2$ by $- 1$ .

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{2 r^{2} \cdot (3 r s) + s^{2}} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 3

Simplify the denominator.

Tap for more steps...

Step 3.1

Factor $s$ out of $2 r^{2} \cdot (3 r s) + s^{2}$ .

Tap for more steps...

Step 3.1.1

Factor $s$ out of $2 r^{2} \cdot (3 r s)$ .

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (2 r^{2} \cdot (3 r)) + s^{2}} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 3.1.2

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

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (2 r^{2} \cdot (3 r)) + s \cdot s} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 3.1.3

Factor $s$ out of $s (2 r^{2} \cdot (3 r)) + s \cdot s$ .

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (2 r^{2} \cdot (3 r) + s)} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 3.2

Rewrite using the commutative property of multiplication.

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (2 \cdot 3 r^{2} r + s)} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 3.3

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

Tap for more steps...

Step 3.3.1

Move $r$ .

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (2 \cdot 3 (r \cdot r^{2}) + s)} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

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

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (2 \cdot 3 (r^{1} r^{2}) + s)} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 3.3.2.2

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

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (2 \cdot 3 r^{1 + 2} + s)} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 3.3.3

Add $1$ and $2$ .

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (2 \cdot 3 r^{3} + s)} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 3.4

Multiply $2$ by $3$ .

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s)} \cdot \frac{4 r^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 4

Factor using the perfect square rule.

Tap for more steps...

Step 4.1

Rewrite $4 r^{2}$ as ${(2 r)}^{2}$ .

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s)} \cdot \frac{{(2 r)}^{2} + 4 r s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 4.2

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

$4 r s = 2 \cdot (2 r) \cdot s$

Step 4.3

Rewrite the polynomial.

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s)} \cdot \frac{{(2 r)}^{2} + 2 \cdot (2 r) \cdot s + s^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 4.4

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

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s)} \cdot \frac{{(2 r + s)}^{2}}{12 r^{2} - 6 r + 3 s^{2}}$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Factor $3$ out of $12 r^{2} - 6 r + 3 s^{2}$ .

Tap for more steps...

Step 5.1.1

Factor $3$ out of $12 r^{2}$ .

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s)} \cdot \frac{{(2 r + s)}^{2}}{3 (4 r^{2}) - 6 r + 3 s^{2}}$

Step 5.1.2

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

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s)} \cdot \frac{{(2 r + s)}^{2}}{3 (4 r^{2}) + 3 (- 2 r) + 3 s^{2}}$

Step 5.1.3

Factor $3$ out of $3 s^{2}$ .

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s)} \cdot \frac{{(2 r + s)}^{2}}{3 (4 r^{2}) + 3 (- 2 r) + 3 (s^{2})}$

Step 5.1.4

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

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s)} \cdot \frac{{(2 r + s)}^{2}}{3 (4 r^{2} - 2 r) + 3 (s^{2})}$

Step 5.1.5

Factor $3$ out of $3 (4 r^{2} - 2 r) + 3 (s^{2})$ .

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s)} \cdot \frac{{(2 r + s)}^{2}}{3 (4 r^{2} - 2 r + s^{2})}$

Step 5.2

Combine.

$\frac{(2 r + s) (4 r^{2} - 2 r s + s^{2}) {(2 r + s)}^{2}}{s (6 r^{3} + s) (3 (4 r^{2} - 2 r + s^{2}))}$

Step 6

Multiply $2 r + s$ by ${(2 r + s)}^{2}$ by adding the exponents.

Tap for more steps...

Step 6.1

Move ${(2 r + s)}^{2}$ .

$\frac{{(2 r + s)}^{2} (2 r + s) (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s) (3 (4 r^{2} - 2 r + s^{2}))}$

Step 6.2

Multiply ${(2 r + s)}^{2}$ by $2 r + s$ .

Tap for more steps...

Step 6.2.1

Raise $2 r + s$ to the power of $1$ .

$\frac{{(2 r + s)}^{2} {(2 r + s)}^{1} (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s) (3 (4 r^{2} - 2 r + s^{2}))}$

Step 6.2.2

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

$\frac{{(2 r + s)}^{2 + 1} (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s) (3 (4 r^{2} - 2 r + s^{2}))}$

Step 6.3

Add $2$ and $1$ .

$\frac{{(2 r + s)}^{3} (4 r^{2} - 2 r s + s^{2})}{s (6 r^{3} + s) (3 (4 r^{2} - 2 r + s^{2}))}$

Step 7

Move $3$ to the left of $s (6 r^{3} + s)$ .

$\frac{{(2 r + s)}^{3} (4 r^{2} - 2 r s + s^{2})}{3 s (6 r^{3} + s) (4 r^{2} - 2 r + s^{2})}$