Algebra Examples

$\frac{3 m^{3} - 3 m}{6 m^{4}} \cdot \frac{4 m^{5}}{m + 1}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Factor $3 m$ out of $3 m^{3}$ .

$\frac{3 m (m^{2}) - 3 m}{6 m^{4}} \cdot \frac{4 m^{5}}{m + 1}$

Step 1.1.2

Factor $3 m$ out of $- 3 m$ .

$\frac{3 m (m^{2}) + 3 m (- 1)}{6 m^{4}} \cdot \frac{4 m^{5}}{m + 1}$

Step 1.1.3

Factor $3 m$ out of $3 m (m^{2}) + 3 m (- 1)$ .

$\frac{3 m (m^{2} - 1)}{6 m^{4}} \cdot \frac{4 m^{5}}{m + 1}$

Step 1.2

Rewrite $1$ as $1^{2}$ .

$\frac{3 m (m^{2} - 1^{2})}{6 m^{4}} \cdot \frac{4 m^{5}}{m + 1}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = m$ and $b = 1$ .

$\frac{3 m (m + 1) (m - 1)}{6 m^{4}} \cdot \frac{4 m^{5}}{m + 1}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $m + 1$ .

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Step 2.1.1

Factor $m + 1$ out of $3 m (m + 1) (m - 1)$ .

$\frac{(m + 1) (3 m (m - 1))}{6 m^{4}} \cdot \frac{4 m^{5}}{m + 1}$

Step 2.1.2

Cancel the common factor.

$\frac{(m + 1) (3 m (m - 1))}{6 m^{4}} \cdot \frac{4 m^{5}}{m + 1}$

Step 2.1.3

Rewrite the expression.

$\frac{3 m (m - 1)}{6 m^{4}} \cdot (4 m^{5})$

Step 2.2

Cancel the common factor of $2 m^{4}$ .

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Step 2.2.1

Factor $2 m^{4}$ out of $6 m^{4}$ .

$\frac{3 m (m - 1)}{2 m^{4} (3)} \cdot (4 m^{5})$

Step 2.2.2

Factor $2 m^{4}$ out of $4 m^{5}$ .

$\frac{3 m (m - 1)}{2 m^{4} (3)} \cdot (2 m^{4} (2 m))$

Step 2.2.3

Cancel the common factor.

$\frac{3 m (m - 1)}{2 m^{4} \cdot 3} \cdot (2 m^{4} (2 m))$

Step 2.2.4

Rewrite the expression.

$\frac{3 m (m - 1)}{3} \cdot (2 m)$

Step 2.3

Combine $2$ and $\frac{3 m (m - 1)}{3}$ .

$\frac{2 (3 m (m - 1))}{3} \cdot m$

Step 2.4

Multiply $3$ by $2$ .

$\frac{6 (m (m - 1))}{3} \cdot m$

Step 2.5

Combine $\frac{6 (m (m - 1))}{3}$ and $m$ .

$\frac{6 (m (m - 1)) m}{3}$

Step 3

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

$\frac{6 (m^{1} m (m - 1))}{3}$

Step 4

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

$\frac{6 (m^{1} m^{1} (m - 1))}{3}$

Step 5

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

$\frac{6 (m^{1 + 1} (m - 1))}{3}$

Step 6

Add $1$ and $1$ .

$\frac{6 (m^{2} (m - 1))}{3}$

Step 7

Cancel the common factor of $6$ and $3$ .

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Step 7.1

Factor $3$ out of $6 (m^{2} (m - 1))$ .

$\frac{3 (2 (m^{2} (m - 1)))}{3}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $3$ out of $3$ .

$\frac{3 (2 (m^{2} (m - 1)))}{3 (1)}$

Step 7.2.2

Cancel the common factor.

$\frac{3 (2 (m^{2} (m - 1)))}{3 \cdot 1}$

Step 7.2.3

Rewrite the expression.

$\frac{2 (m^{2} (m - 1))}{1}$

Step 7.2.4

Divide $2 (m^{2} (m - 1))$ by $1$ .

$2 (m^{2} (m - 1))$

Step 8

Apply the distributive property.

$2 (m^{2} m + m^{2} \cdot - 1)$

Step 9

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

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Step 9.1

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

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Step 9.1.1

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

$2 (m^{2} m^{1} + m^{2} \cdot - 1)$

Step 9.1.2

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

$2 (m^{2 + 1} + m^{2} \cdot - 1)$

Step 9.2

Add $2$ and $1$ .

$2 (m^{3} + m^{2} \cdot - 1)$

Step 10

Move $- 1$ to the left of $m^{2}$ .

$2 (m^{3} - 1 \cdot m^{2})$

Step 11

Rewrite $- 1 m^{2}$ as $- m^{2}$ .

$2 (m^{3} - m^{2})$

Step 12

Simplify by multiplying through.

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Step 12.1

Apply the distributive property.

$2 m^{3} + 2 (- m^{2})$

Step 12.2

Multiply $- 1$ by $2$ .

$2 m^{3} - 2 m^{2}$