Algebra Examples

$\frac{2 r^{3} s^{8} (7 r^{7} - 5 r^{3} s^{4}) - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the distributive property.

$\frac{2 r^{3} s^{8} (7 r^{7}) + 2 r^{3} s^{8} (- 5 r^{3} s^{4}) - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.2

Multiply $r^{3}$ by $r^{7}$ by adding the exponents.

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Step 1.2.1

Move $r^{7}$ .

$\frac{2 (r^{7} r^{3}) s^{8} \cdot 7 + 2 r^{3} s^{8} (- 5 r^{3} s^{4}) - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.2.2

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

$\frac{2 r^{7 + 3} s^{8} \cdot 7 + 2 r^{3} s^{8} (- 5 r^{3} s^{4}) - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.2.3

Add $7$ and $3$ .

$\frac{2 r^{10} s^{8} \cdot 7 + 2 r^{3} s^{8} (- 5 r^{3} s^{4}) - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.3

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

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Step 1.3.1

Move $r^{3}$ .

$\frac{2 r^{10} s^{8} \cdot 7 + 2 (r^{3} r^{3}) s^{8} (- 5 s^{4}) - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.3.2

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

$\frac{2 r^{10} s^{8} \cdot 7 + 2 r^{3 + 3} s^{8} (- 5 s^{4}) - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.3.3

Add $3$ and $3$ .

$\frac{2 r^{10} s^{8} \cdot 7 + 2 r^{6} s^{8} (- 5 s^{4}) - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $7$ by $2$ .

$\frac{14 r^{10} s^{8} + 2 r^{6} s^{8} (- 5 s^{4}) - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.4.2

Multiply $s^{8}$ by $s^{4}$ by adding the exponents.

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Step 1.4.2.1

Move $s^{4}$ .

$\frac{14 r^{10} s^{8} + 2 r^{6} (s^{4} s^{8}) \cdot - 5 - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.4.2.2

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

$\frac{14 r^{10} s^{8} + 2 r^{6} s^{4 + 8} \cdot - 5 - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.4.2.3

Add $4$ and $8$ .

$\frac{14 r^{10} s^{8} + 2 r^{6} s^{12} \cdot - 5 - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.4.3

Multiply $- 5$ by $2$ .

$\frac{14 r^{10} s^{8} - 10 r^{6} s^{12} - (2 r^{10} s^{8} - 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.5

Apply the distributive property.

$\frac{14 r^{10} s^{8} - 10 r^{6} s^{12} - (2 r^{10} s^{8}) - (- 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.6

Multiply $2$ by $- 1$ .

$\frac{14 r^{10} s^{8} - 10 r^{6} s^{12} - 2 (r^{10} s^{8}) - (- 6 r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.7

Multiply $- 6$ by $- 1$ .

$\frac{14 r^{10} s^{8} - 10 r^{6} s^{12} - 2 (r^{10} s^{8}) + 6 (r^{6} s^{12})}{4 r^{2} s^{4}}$

Step 1.8

Remove parentheses.

$\frac{14 r^{10} s^{8} - 10 r^{6} s^{12} - 2 r^{10} s^{8} + 6 r^{6} s^{12}}{4 r^{2} s^{4}}$

Step 1.9

Subtract $2 r^{10} s^{8}$ from $14 r^{10} s^{8}$ .

$\frac{12 r^{10} s^{8} - 10 r^{6} s^{12} + 6 r^{6} s^{12}}{4 r^{2} s^{4}}$

Step 1.10

Add $- 10 r^{6} s^{12}$ and $6 r^{6} s^{12}$ .

$\frac{12 r^{10} s^{8} - 4 r^{6} s^{12}}{4 r^{2} s^{4}}$

Step 1.11

Rewrite $12 r^{10} s^{8} - 4 r^{6} s^{12}$ in a factored form.

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Step 1.11.1

Factor $4 r^{6} s^{8}$ out of $12 r^{10} s^{8} - 4 r^{6} s^{12}$ .

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Step 1.11.1.1

Factor $4 r^{6} s^{8}$ out of $12 r^{10} s^{8}$ .

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

Step 1.11.1.2

Factor $4 r^{6} s^{8}$ out of $- 4 r^{6} s^{12}$ .

$\frac{4 r^{6} s^{8} (3 r^{4}) + 4 r^{6} s^{8} (- 1 s^{4})}{4 r^{2} s^{4}}$

Step 1.11.1.3

Factor $4 r^{6} s^{8}$ out of $4 r^{6} s^{8} (3 r^{4}) + 4 r^{6} s^{8} (- 1 s^{4})$ .

$\frac{4 r^{6} s^{8} (3 r^{4} - 1 s^{4})}{4 r^{2} s^{4}}$

Step 1.11.2

Rewrite $- 1 s^{4}$ as $- s^{4}$ .

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

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $4$ .

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

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

Step 2.2

Simplify terms.

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Step 2.2.1

Cancel the common factor of $r^{6}$ and $r^{2}$ .

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Step 2.2.1.1

Factor $r^{2}$ out of $r^{6} s^{8} (3 r^{4} - s^{4})$ .

$\frac{r^{2} (r^{4} s^{8} (3 r^{4} - s^{4}))}{r^{2} s^{4}}$

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Factor $r^{2}$ out of $r^{2} s^{4}$ .

$\frac{r^{2} (r^{4} s^{8} (3 r^{4} - s^{4}))}{r^{2} (s^{4})}$

Step 2.2.1.2.2

Cancel the common factor.

$\frac{r^{2} (r^{4} s^{8} (3 r^{4} - s^{4}))}{r^{2} s^{4}}$

Step 2.2.1.2.3

Rewrite the expression.

$\frac{r^{4} s^{8} (3 r^{4} - s^{4})}{s^{4}}$

Step 2.2.2

Cancel the common factor of $s^{8}$ and $s^{4}$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.2.2.2.4

Divide $r^{4} s^{4} (3 r^{4} - s^{4})$ by $1$ .

$r^{4} s^{4} (3 r^{4} - s^{4})$

Step 2.2.3

Apply the distributive property.

$r^{4} s^{4} (3 r^{4}) + r^{4} s^{4} (- s^{4})$

Step 2.2.4

Reorder.

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Step 2.2.4.1

Rewrite using the commutative property of multiplication.

$3 r^{4} s^{4} r^{4} + r^{4} s^{4} (- s^{4})$

Step 2.2.4.2

Rewrite using the commutative property of multiplication.

$3 r^{4} s^{4} r^{4} - r^{4} s^{4} s^{4}$

Step 2.3

Simplify each term.

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Step 2.3.1

Multiply $r^{4}$ by $r^{4}$ by adding the exponents.

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Step 2.3.1.1

Move $r^{4}$ .

$3 (r^{4} r^{4}) s^{4} - r^{4} s^{4} s^{4}$

Step 2.3.1.2

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

$3 r^{4 + 4} s^{4} - r^{4} s^{4} s^{4}$

Step 2.3.1.3

Add $4$ and $4$ .

$3 r^{8} s^{4} - r^{4} s^{4} s^{4}$

Step 2.3.2

Multiply $s^{4}$ by $s^{4}$ by adding the exponents.

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Step 2.3.2.1

Move $s^{4}$ .

$3 r^{8} s^{4} - r^{4} (s^{4} s^{4})$

Step 2.3.2.2

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

$3 r^{8} s^{4} - r^{4} s^{4 + 4}$

Step 2.3.2.3

Add $4$ and $4$ .

$3 r^{8} s^{4} - r^{4} s^{8}$