Simplify square root of r^2-1(1+(r^2)/(r^2-1))+2r-(1+r/( square root of r^2-1))/(r+ square root of r^2-1)

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

Step 1

Simplify each term.

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

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

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

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

Simplify the denominator.

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

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Expand $(r + 1) (r - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $r$ by $r$ .

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

Move $- 1$ to the left of $r$ .

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

Rewrite $- 1 r$ as $- r$ .

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

Multiply $r$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Add $- r$ and $r$ .

$\sqrt{(r + 1) (r - 1)} \frac{r^{2} + 0 - 1 + r^{2}}{(r + 1) (r - 1)} + 2 r - \frac{1 + \frac{r}{\sqrt{r^{2} - 1}}}{r + \sqrt{r^{2} - 1}}$

Add $r^{2}$ and $0$ .

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

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

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

Combine $\sqrt{(r + 1) (r - 1)}$ and $\frac{2 r^{2} - 1}{(r + 1) (r - 1)}$ .

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

Multiply the numerator and denominator of the complex fraction by $\sqrt{r^{2} - 1}$ .

Multiply $\frac{1 + \frac{r}{\sqrt{r^{2} - 1}}}{r + \sqrt{r^{2} - 1}}$ by $\frac{\sqrt{r^{2} - 1}}{\sqrt{r^{2} - 1}}$ .

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

Combine.

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

Apply the distributive property.

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

Cancel the common factor of $\sqrt{r^{2} - 1}$ .

Cancel the common factor.

Rewrite the expression.

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

Simplify the numerator.

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

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

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

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

Multiply $\sqrt{(r + 1) (r - 1)}$ by $1$ .

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

Simplify the denominator.

Factor $\sqrt{r^{2} - 1}$ out of $\sqrt{r^{2} - 1} r + \sqrt{r^{2} - 1} \sqrt{r^{2} - 1}$ .

Factor $\sqrt{r^{2} - 1}$ out of $\sqrt{r^{2} - 1} r$ .

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

Factor $\sqrt{r^{2} - 1}$ out of $\sqrt{r^{2} - 1} \sqrt{r^{2} - 1}$ .

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

Factor $\sqrt{r^{2} - 1}$ out of $\sqrt{r^{2} - 1} (r) + \sqrt{r^{2} - 1} (\sqrt{r^{2} - 1})$ .

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

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

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

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

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

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

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

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

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

Cancel the common factor of $\sqrt{(r + 1) (r - 1)} + r$ and $r + \sqrt{(r + 1) (r - 1)}$ .

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{1}{\sqrt{(r + 1) (r - 1)}}$ by $\frac{\sqrt{(r + 1) (r - 1)}}{\sqrt{(r + 1) (r - 1)}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{(r + 1) (r - 1)}}$ by $\frac{\sqrt{(r + 1) (r - 1)}}{\sqrt{(r + 1) (r - 1)}}$ .

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

Raise $\sqrt{(r + 1) (r - 1)}$ to the power of $1$ .

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

Raise $\sqrt{(r + 1) (r - 1)}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{(r + 1) (r - 1)}}^{2}$ as $(r + 1) (r - 1)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(r + 1) (r - 1)}$ as ${((r + 1) (r - 1))}^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

Apply the distributive property.

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

Rewrite using the commutative property of multiplication.

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

Move $- 1$ to the left of $\sqrt{(r + 1) (r - 1)}$ .

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

Rewrite $- 1 \sqrt{(r + 1) (r - 1)}$ as $- \sqrt{(r + 1) (r - 1)}$ .

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

Step 4

Subtract $\sqrt{(r + 1) (r - 1)}$ from $- \sqrt{(r + 1) (r - 1)}$ .

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

Step 5

Simplify each term.

Simplify the numerator.

Factor $2 \sqrt{(r + 1) (r - 1)}$ out of $2 \sqrt{(r + 1) (r - 1)} r^{2} - 2 \sqrt{(r + 1) (r - 1)}$ .

Factor $2 \sqrt{(r + 1) (r - 1)}$ out of $2 \sqrt{(r + 1) (r - 1)} r^{2}$ .

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

Factor $2 \sqrt{(r + 1) (r - 1)}$ out of $- 2 \sqrt{(r + 1) (r - 1)}$ .

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

Factor $2 \sqrt{(r + 1) (r - 1)}$ out of $2 \sqrt{(r + 1) (r - 1)} (r^{2}) + 2 \sqrt{(r + 1) (r - 1)} (- 1)$ .

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

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

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

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

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

Cancel the common factor of $r + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $r - 1$ .

Cancel the common factor.

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

Divide $2 \sqrt{(r + 1) (r - 1)}$ by $1$ .

$2 r + 2 \sqrt{(r + 1) (r - 1)}$

Simplify square root of r^2-1(1+(r^2)/(r^2-1))+2r-(1+r/( square root of r^2-1))/(r+ square root of r^2-1)

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