Finite Math Examples

$\frac{e (3 m^{2} + 15 m)}{m^{2}} \div \frac{30 m}{m^{2} + 5 m}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{e (3 m^{2} + 15 m)}{m^{2}} \cdot \frac{m^{2} + 5 m}{30 m}$

Step 2

Factor $3 m$ out of $3 m^{2} + 15 m$ .

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Step 2.1

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

$\frac{e (3 m (m) + 15 m)}{m^{2}} \cdot \frac{m^{2} + 5 m}{30 m}$

Step 2.2

Factor $3 m$ out of $15 m$ .

$\frac{e (3 m (m) + 3 m (5))}{m^{2}} \cdot \frac{m^{2} + 5 m}{30 m}$

Step 2.3

Factor $3 m$ out of $3 m (m) + 3 m (5)$ .

$\frac{e (3 m (m + 5))}{m^{2}} \cdot \frac{m^{2} + 5 m}{30 m}$

$\frac{e \cdot 3 m (m + 5)}{m^{2}} \cdot \frac{m^{2} + 5 m}{30 m}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $3 m$ .

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Step 3.1.1

Factor $3 m$ out of $e \cdot 3 m (m + 5)$ .

$\frac{3 m (e (m + 5))}{m^{2}} \cdot \frac{m^{2} + 5 m}{30 m}$

Step 3.1.2

Factor $3 m$ out of $30 m$ .

$\frac{3 m (e (m + 5))}{m^{2}} \cdot \frac{m^{2} + 5 m}{3 m (10)}$

Step 3.1.3

Cancel the common factor.

$\frac{3 m (e (m + 5))}{m^{2}} \cdot \frac{m^{2} + 5 m}{3 m \cdot 10}$

Step 3.1.4

Rewrite the expression.

$\frac{e (m + 5)}{m^{2}} \cdot \frac{m^{2} + 5 m}{10}$

Step 3.2

Multiply $\frac{e (m + 5)}{m^{2}}$ by $\frac{m^{2} + 5 m}{10}$ .

$\frac{e (m + 5) (m^{2} + 5 m)}{m^{2} \cdot 10}$

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

$\frac{e (m + 5) (m \cdot m + 5 m)}{m^{2} \cdot 10}$

Step 4.1.2

Factor $m$ out of $5 m$ .

$\frac{e (m + 5) (m \cdot m + m \cdot 5)}{m^{2} \cdot 10}$

Step 4.1.3

Factor $m$ out of $m \cdot m + m \cdot 5$ .

$\frac{e (m + 5) (m (m + 5))}{m^{2} \cdot 10}$

$\frac{e (m + 5) m (m + 5)}{m^{2} \cdot 10}$

Step 4.2

Combine exponents.

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Step 4.2.1

Raise $m + 5$ to the power of $1$ .

$\frac{e m ({(m + 5)}^{1} (m + 5))}{m^{2} \cdot 10}$

Step 4.2.2

Raise $m + 5$ to the power of $1$ .

$\frac{e m ({(m + 5)}^{1} {(m + 5)}^{1})}{m^{2} \cdot 10}$

Step 4.2.3

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

$\frac{e m {(m + 5)}^{1 + 1}}{m^{2} \cdot 10}$

Step 4.2.4

Add $1$ and $1$ .

$\frac{e m {(m + 5)}^{2}}{m^{2} \cdot 10}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

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

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Step 5.1.1

Factor $m$ out of $e m {(m + 5)}^{2}$ .

$\frac{m (e {(m + 5)}^{2})}{m^{2} \cdot 10}$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Factor $m$ out of $m^{2} \cdot 10$ .

$\frac{m (e {(m + 5)}^{2})}{m (m \cdot 10)}$

Step 5.1.2.2

Cancel the common factor.

$\frac{m (e {(m + 5)}^{2})}{m (m \cdot 10)}$

Step 5.1.2.3

Rewrite the expression.

$\frac{e {(m + 5)}^{2}}{m \cdot 10}$

Step 5.2

Move $10$ to the left of $m$ .

$\frac{e {(m + 5)}^{2}}{10 m}$