Basic Math Examples

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

Step 1

To write $m$ as a fraction with a common denominator, multiply by $\frac{2 m}{2 m}$ .

$m \cdot \frac{2 m}{2 m} + \frac{3}{2 m} + m - \frac{2}{m}$

Step 2

Simplify terms.

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Step 2.1

Combine $m$ and $\frac{2 m}{2 m}$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

Rewrite using the commutative property of multiplication.

$\frac{2 m \cdot m + 3}{2 m} + m - \frac{2}{m}$

Step 3.2

Multiply $m$ by $m$ by adding the exponents.

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Step 3.2.1

Move $m$ .

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

Step 3.2.2

Multiply $m$ by $m$ .

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

Step 4

To write $m$ as a fraction with a common denominator, multiply by $\frac{2 m}{2 m}$ .

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

Step 5

Simplify terms.

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Step 5.1

Combine $m$ and $\frac{2 m}{2 m}$ .

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Rewrite using the commutative property of multiplication.

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

Step 6.2

Multiply $m$ by $m$ by adding the exponents.

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Step 6.2.1

Move $m$ .

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

Step 6.2.2

Multiply $m$ by $m$ .

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

Step 6.3

Add $2 m^{2}$ and $2 m^{2}$ .

$\frac{4 m^{2} + 3}{2 m} - \frac{2}{m}$

Step 7

To write $- \frac{2}{m}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4 m^{2} + 3}{2 m} - \frac{2}{m} \cdot \frac{2}{2}$

Step 8

Write each expression with a common denominator of $2 m$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{2}{m}$ by $\frac{2}{2}$ .

$\frac{4 m^{2} + 3}{2 m} - \frac{2 \cdot 2}{m \cdot 2}$

Step 8.2

Reorder the factors of $m \cdot 2$ .

$\frac{4 m^{2} + 3}{2 m} - \frac{2 \cdot 2}{2 m}$

Step 9

Combine the numerators over the common denominator.

$\frac{4 m^{2} + 3 - 2 \cdot 2}{2 m}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 2$ by $2$ .

$\frac{4 m^{2} + 3 - 4}{2 m}$

Step 10.2

Subtract $4$ from $3$ .

$\frac{4 m^{2} - 1}{2 m}$

Step 10.3

Rewrite $4 m^{2} - 1$ in a factored form.

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Step 10.3.1

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

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

Step 10.3.2

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

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

Step 10.3.3

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

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