Basic Math Examples

$\frac{\frac{1}{r + 2} - 3}{\frac{4}{r} - r}$

Step 1

Multiply the numerator and denominator of the fraction by $(r + 2) r$ .

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Step 1.1

Multiply $\frac{\frac{1}{r + 2} - 3}{\frac{4}{r} - r}$ by $\frac{(r + 2) r}{(r + 2) r}$ .

$\frac{(r + 2) r}{(r + 2) r} \cdot \frac{\frac{1}{r + 2} - 3}{\frac{4}{r} - r}$

Step 1.2

Combine.

$\frac{(r + 2) r (\frac{1}{r + 2} - 3)}{(r + 2) r (\frac{4}{r} - r)}$

Step 2

Apply the distributive property.

$\frac{(r + 2) r \frac{1}{r + 2} + (r + 2) r \cdot - 3}{(r + 2) r \frac{4}{r} + (r + 2) r (- r)}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $r + 2$ .

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Step 3.1.1

Factor $r + 2$ out of $(r + 2) r$ .

$\frac{(r + 2) (r) \frac{1}{r + 2} + (r + 2) r \cdot - 3}{(r + 2) r \frac{4}{r} + (r + 2) r (- r)}$

Step 3.1.2

Cancel the common factor.

$\frac{(r + 2) r \frac{1}{r + 2} + (r + 2) r \cdot - 3}{(r + 2) r \frac{4}{r} + (r + 2) r (- r)}$

Step 3.1.3

Rewrite the expression.

$\frac{r + (r + 2) r \cdot - 3}{(r + 2) r \frac{4}{r} + (r + 2) r (- r)}$

Step 3.2

Cancel the common factor of $r$ .

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Step 3.2.1

Factor $r$ out of $(r + 2) r$ .

$\frac{r + (r + 2) r \cdot - 3}{r (r + 2) \frac{4}{r} + (r + 2) r (- r)}$

Step 3.2.2

Cancel the common factor.

$\frac{r + (r + 2) r \cdot - 3}{r (r + 2) \frac{4}{r} + (r + 2) r (- r)}$

Step 3.2.3

Rewrite the expression.

$\frac{r + (r + 2) r \cdot - 3}{(r + 2) \cdot 4 + (r + 2) r (- r)}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $r$ out of $r + (r + 2) r \cdot - 3$ .

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Step 4.1.1

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

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

Step 4.1.2

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

$\frac{r \cdot 1 + (r + 2) r \cdot - 3}{(r + 2) \cdot 4 + (r + 2) r (- r)}$

Step 4.1.3

Factor $r$ out of $(r + 2) r \cdot - 3$ .

$\frac{r \cdot 1 + r ((r + 2) \cdot - 3)}{(r + 2) \cdot 4 + (r + 2) r (- r)}$

Step 4.1.4

Factor $r$ out of $r \cdot 1 + r ((r + 2) \cdot - 3)$ .

$\frac{r (1 + (r + 2) \cdot - 3)}{(r + 2) \cdot 4 + (r + 2) r (- r)}$

Step 4.2

Apply the distributive property.

$\frac{r (1 + r \cdot - 3 + 2 \cdot - 3)}{(r + 2) \cdot 4 + (r + 2) r (- r)}$

Step 4.3

Move $- 3$ to the left of $r$ .

$\frac{r (1 - 3 \cdot r + 2 \cdot - 3)}{(r + 2) \cdot 4 + (r + 2) r (- r)}$

Step 4.4

Multiply $2$ by $- 3$ .

$\frac{r (1 - 3 \cdot r - 6)}{(r + 2) \cdot 4 + (r + 2) r (- r)}$

Step 4.5

Subtract $6$ from $1$ .

$\frac{r (- 3 r - 5)}{(r + 2) \cdot 4 + (r + 2) r (- r)}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $r + 2$ out of $(r + 2) \cdot 4 + (r + 2) r (- r)$ .

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Step 5.1.1

Factor $r + 2$ out of $(r + 2) \cdot 4$ .

$\frac{r (- 3 r - 5)}{(r + 2) (4) + (r + 2) r (- r)}$

Step 5.1.2

Factor $r + 2$ out of $(r + 2) r (- r)$ .

$\frac{r (- 3 r - 5)}{(r + 2) (4) + (r + 2) (r (- r))}$

Step 5.1.3

Factor $r + 2$ out of $(r + 2) (4) + (r + 2) (r (- r))$ .

$\frac{r (- 3 r - 5)}{(r + 2) (4 + r (- r))}$

Step 5.2

Rewrite $4$ as $2^{2}$ .

$\frac{r (- 3 r - 5)}{(r + 2) (2^{2} + r (- r))}$

Step 5.3

Rewrite $r (r)$ as $r^{2}$ .

$\frac{r (- 3 r - 5)}{(r + 2) (2^{2} - r^{2})}$

Step 5.4

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

$\frac{r (- 3 r - 5)}{(r + 2) (2 + r) (2 - r)}$

Step 5.5

Combine exponents.

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Step 5.5.1

Reorder terms.

$\frac{r (- 3 r - 5)}{(r + 2) (r + 2) (2 - r)}$

Step 5.5.2

Raise $r + 2$ to the power of $1$ .

$\frac{r (- 3 r - 5)}{{(r + 2)}^{1} (r + 2) (2 - r)}$

Step 5.5.3

Raise $r + 2$ to the power of $1$ .

$\frac{r (- 3 r - 5)}{{(r + 2)}^{1} {(r + 2)}^{1} (2 - r)}$

Step 5.5.4

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

$\frac{r (- 3 r - 5)}{{(r + 2)}^{1 + 1} (2 - r)}$

Step 5.5.5

Add $1$ and $1$ .

$\frac{r (- 3 r - 5)}{{(r + 2)}^{2} (2 - r)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 3 r$ .

$\frac{r (- (3 r) - 5)}{{(r + 2)}^{2} (2 - r)}$

Step 6.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{r (- (3 r) - 1 (5))}{{(r + 2)}^{2} (2 - r)}$

Step 6.3

Factor $- 1$ out of $- (3 r) - 1 (5)$ .

$\frac{r (- (3 r + 5))}{{(r + 2)}^{2} (2 - r)}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (3 r + 5)$ as $- 1 (3 r + 5)$ .

$\frac{r (- 1 (3 r + 5))}{{(r + 2)}^{2} (2 - r)}$

Step 6.4.2

Move the negative in front of the fraction.

$- \frac{r (3 r + 5)}{{(r + 2)}^{2} (2 - r)}$