Basic Math Examples

$\frac{\frac{1}{m} + 1}{\frac{3}{m^{2} - 2 m} + \frac{1}{m}}$

Step 1

Multiply the numerator and denominator of the fraction by $m (m^{2} - 2 m)$ .

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Step 1.1

Multiply $\frac{\frac{1}{m} + 1}{\frac{3}{m^{2} - 2 m} + \frac{1}{m}}$ by $\frac{m (m^{2} - 2 m)}{m (m^{2} - 2 m)}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $m$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $m$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Factor $m$ out of $- 2 m$ .

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

Step 4.1.3

Factor $m$ out of $m (m^{2} - 2 m) \cdot 1$ .

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

Step 4.1.4

Factor $m$ out of $m \cdot m + m \cdot - 2$ .

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

Step 4.1.5

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

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

Step 4.2

Multiply $m^{2} - 2 m$ by $1$ .

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

Step 4.3

Subtract $2 m$ from $m$ .

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

Step 4.4

Reorder terms.

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

Step 4.5

Factor $m^{2} - m - 2$ using the AC method.

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Step 4.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 4.5.2

Write the factored form using these integers.

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

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

Step 5

Simplify the denominator.

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Step 5.1

Subtract $2 m$ from $m \cdot 3$ .

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Step 5.1.1

Reorder $m$ and $3$ .

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

Step 5.1.2

Subtract $2 m$ from $3 \cdot m$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

Factor $m$ out of $m^{1}$ .

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

Step 5.2.4

Factor $m$ out of $m \cdot m + m \cdot 1$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $m$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $m + 1$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Divide $m - 2$ by $1$ .

$m - 2$