Basic Math Examples

$\frac{r f \cdot \frac{1 + r f}{1 + k f} + r f \cdot (k e l)}{1 - (1 + k f \cdot (r f))}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $r f$ out of $r f \cdot \frac{1 + r f}{1 + k f} + r f \cdot k e l$ .

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Step 1.1.1

Factor $r f$ out of $r f \cdot \frac{1 + r f}{1 + k f}$ .

$\frac{r f (\frac{1 + r f}{1 + k f}) + r f \cdot k e l}{1 - (1 + k f \cdot (r f))}$

Step 1.1.2

Factor $r f$ out of $r f \cdot k e l$ .

$\frac{r f (\frac{1 + r f}{1 + k f}) + r f (k e l)}{1 - (1 + k f \cdot (r f))}$

Step 1.1.3

Factor $r f$ out of $r f (\frac{1 + r f}{1 + k f}) + r f (k e l)$ .

$\frac{r f (\frac{1 + r f}{1 + k f} + k e l)}{1 - (1 + k f \cdot (r f))}$

Step 1.2

To write $k e l$ as a fraction with a common denominator, multiply by $\frac{1 + k f}{1 + k f}$ .

$\frac{r f (\frac{1 + r f}{1 + k f} + \frac{k e l (1 + k f)}{1 + k f})}{1 - (1 + k f \cdot (r f))}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{r f \frac{1 + r f + k e l (1 + k f)}{1 + k f}}{1 - (1 + k f \cdot (r f))}$

Step 1.4

Rewrite $\frac{1 + r f + k e l (1 + k f)}{1 + k f}$ in a factored form.

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Step 1.4.1

Apply the distributive property.

$\frac{r f \frac{1 + r f + k e l \cdot 1 + k e l (k f)}{1 + k f}}{1 - (1 + k f \cdot (r f))}$

Step 1.4.2

Multiply $k$ by $1$ .

$\frac{r f \frac{1 + r f + k e l + k e l (k f)}{1 + k f}}{1 - (1 + k f \cdot (r f))}$

Step 1.4.3

Multiply $k$ by $k$ by adding the exponents.

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Step 1.4.3.1

Move $k$ .

$\frac{r f \frac{1 + r f + k e l + k \cdot k e l f}{1 + k f}}{1 - (1 + k f \cdot (r f))}$

Step 1.4.3.2

Multiply $k$ by $k$ .

$\frac{r f \frac{1 + r f + k e l + k^{2} e l f}{1 + k f}}{1 - (1 + k f \cdot (r f))}$

Step 1.5

Combine exponents.

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Step 1.5.1

Combine $r$ and $\frac{1 + r f + k e l + k^{2} e l f}{1 + k f}$ .

$\frac{f \frac{r (1 + r f + k e l + k^{2} e l f)}{1 + k f}}{1 - (1 + k f \cdot (r f))}$

Step 1.5.2

Combine $f$ and $\frac{r (1 + r f + k e l + k^{2} e l f)}{1 + k f}$ .

$\frac{\frac{f (r (1 + r f + k e l + k^{2} e l f))}{1 + k f}}{1 - (1 + k f \cdot (r f))}$

Step 1.6

Remove unnecessary parentheses.

$\frac{\frac{f r (1 + r f + k e l + k^{2} e l f)}{1 + k f}}{1 - (1 + k f \cdot (r f))}$

Step 2

Simplify the denominator.

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Step 2.1

Multiply $f$ by $f$ by adding the exponents.

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Step 2.1.1

Move $f$ .

$\frac{\frac{f r (1 + r f + k e l + k^{2} e l f)}{1 + k f}}{1 - (1 + k (f \cdot f) \cdot r)}$

Step 2.1.2

Multiply $f$ by $f$ .

$\frac{\frac{f r (1 + r f + k e l + k^{2} e l f)}{1 + k f}}{1 - (1 + k f^{2} \cdot r)}$

$\frac{\frac{f r (1 + r f + k e l + k^{2} e l f)}{1 + k f}}{1 - (1 + k f^{2} r)}$

Step 2.2

Apply the distributive property.

$\frac{\frac{f r (1 + r f + k e l + k^{2} e l f)}{1 + k f}}{1 - 1 \cdot 1 - (k f^{2} r)}$

Step 2.3

Multiply $- 1$ by $1$ .

$\frac{\frac{f r (1 + r f + k e l + k^{2} e l f)}{1 + k f}}{1 - 1 - (k f^{2} r)}$

Step 2.4

Subtract $1$ from $1$ .

$\frac{\frac{f r (1 + r f + k e l + k^{2} e l f)}{1 + k f}}{0 - k f^{2} r}$

Step 2.5

Subtract $k f^{2} r$ from $0$ .

$\frac{\frac{f r (1 + r f + k e l + k^{2} e l f)}{1 + k f}}{- k f^{2} r}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{f r (1 + r f + k e l + k^{2} e l f)}{1 + k f} \cdot \frac{1}{- k f^{2} r}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $f r$ .

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Step 4.1.1

Factor $f r$ out of $- k f^{2} r$ .

$\frac{f r (1 + r f + k e l + k^{2} e l f)}{1 + k f} \cdot \frac{1}{f r (- k f)}$

Step 4.1.2

Cancel the common factor.

$\frac{f r (1 + r f + k e l + k^{2} e l f)}{1 + k f} \cdot \frac{1}{f r (- k f)}$

Step 4.1.3

Rewrite the expression.

$\frac{1 + r f + k e l + k^{2} e l f}{1 + k f} \cdot \frac{1}{- k f}$

Step 4.2

Multiply $\frac{1 + r f + k e l + k^{2} e l f}{1 + k f}$ by $\frac{1}{- k f}$ .

$\frac{1 + r f + k e l + k^{2} e l f}{(1 + k f) (- k f)}$

Step 5

Factor out negative.

$\frac{1 + r f + k e l + k^{2} e l f}{- (1 + k f) k f}$

Step 6

Rewrite negatives.

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Step 6.1

Move the negative in front of the fraction.

$- \frac{1 + r f + k e l + k^{2} e l f}{(1 + k f) k f}$

Step 6.2

Reorder factors in $- \frac{1 + r f + k e l + k^{2} e l f}{(1 + k f) k f}$ .

$- \frac{1 + r f + k e l + k^{2} e l f}{k f (1 + k f)}$