Simplify ( square root of x- square root of x+1)/( square root of x+ square root of x+1)

$\frac{\sqrt{x} - \sqrt{x + 1}}{\sqrt{x} + \sqrt{x + 1}}$

Multiply $\frac{\sqrt{x} - \sqrt{x + 1}}{\sqrt{x} + \sqrt{x + 1}}$ by $\frac{\sqrt{x} - \sqrt{x + 1}}{\sqrt{x} - \sqrt{x + 1}}$ .

$\frac{\sqrt{x} - \sqrt{x + 1}}{\sqrt{x} + \sqrt{x + 1}} \cdot \frac{\sqrt{x} - \sqrt{x + 1}}{\sqrt{x} - \sqrt{x + 1}}$

Combine fractions.

Multiply $\frac{\sqrt{x} - \sqrt{x + 1}}{\sqrt{x} + \sqrt{x + 1}}$ by $\frac{\sqrt{x} - \sqrt{x + 1}}{\sqrt{x} - \sqrt{x + 1}}$ .

$\frac{(\sqrt{x} - \sqrt{x + 1}) (\sqrt{x} - \sqrt{x + 1})}{(\sqrt{x} + \sqrt{x + 1}) (\sqrt{x} - \sqrt{x + 1})}$

Expand the denominator using the FOIL method.

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

Simplify.

$\frac{(\sqrt{x} - \sqrt{x + 1}) (\sqrt{x} - \sqrt{x + 1})}{- 1}$

Simplify the expression.

Move the negative one from the denominator of $\frac{(\sqrt{x} - \sqrt{x + 1}) (\sqrt{x} - \sqrt{x + 1})}{- 1}$ .

$- 1 \cdot ((\sqrt{x} - \sqrt{x + 1}) (\sqrt{x} - \sqrt{x + 1}))$

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

$- ((\sqrt{x} - \sqrt{x + 1}) (\sqrt{x} - \sqrt{x + 1}))$

Expand $(\sqrt{x} - \sqrt{x + 1}) (\sqrt{x} - \sqrt{x + 1})$ using the FOIL Method.

Apply the distributive property.

$- (\sqrt{x} (\sqrt{x} - \sqrt{x + 1}) - \sqrt{x + 1} (\sqrt{x} - \sqrt{x + 1}))$

Apply the distributive property.

$- (\sqrt{x} \sqrt{x} + \sqrt{x} (- \sqrt{x + 1}) - \sqrt{x + 1} (\sqrt{x} - \sqrt{x + 1}))$

Apply the distributive property.

$- (\sqrt{x} \sqrt{x} + \sqrt{x} (- \sqrt{x + 1}) - \sqrt{x + 1} \sqrt{x} - \sqrt{x + 1} (- \sqrt{x + 1}))$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x} \sqrt{x}$ .

Raise $\sqrt{x}$ to the power of $1$ .

$- ({\sqrt{x}}^{1} \sqrt{x} + \sqrt{x} (- \sqrt{x + 1}) - \sqrt{x + 1} \sqrt{x} - \sqrt{x + 1} (- \sqrt{x + 1}))$

Raise $\sqrt{x}$ to the power of $1$ .

$- ({\sqrt{x}}^{1} {\sqrt{x}}^{1} + \sqrt{x} (- \sqrt{x + 1}) - \sqrt{x + 1} \sqrt{x} - \sqrt{x + 1} (- \sqrt{x + 1}))$

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

$- ({\sqrt{x}}^{1 + 1} + \sqrt{x} (- \sqrt{x + 1}) - \sqrt{x + 1} \sqrt{x} - \sqrt{x + 1} (- \sqrt{x + 1}))$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$- (x^{1} + \sqrt{x} (- \sqrt{x + 1}) - \sqrt{x + 1} \sqrt{x} - \sqrt{x + 1} (- \sqrt{x + 1}))$

Simplify.

$- (x + \sqrt{x} (- \sqrt{x + 1}) - \sqrt{x + 1} \sqrt{x} - \sqrt{x + 1} (- \sqrt{x + 1}))$

Rewrite using the commutative property of multiplication.

$- (x - \sqrt{x} \sqrt{x + 1} - \sqrt{x + 1} \sqrt{x} - \sqrt{x + 1} (- \sqrt{x + 1}))$

Combine using the product rule for radicals.

$- (x - \sqrt{(x + 1) x} - \sqrt{x + 1} \sqrt{x} - \sqrt{x + 1} (- \sqrt{x + 1}))$

Combine using the product rule for radicals.

$- (x - \sqrt{(x + 1) x} - \sqrt{x (x + 1)} - \sqrt{x + 1} (- \sqrt{x + 1}))$

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

Multiply $- 1$ by $- 1$ .

$- (x - \sqrt{(x + 1) x} - \sqrt{x (x + 1)} + 1 \sqrt{x + 1} \sqrt{x + 1})$

Multiply $\sqrt{x + 1}$ by $1$ .

$- (x - \sqrt{(x + 1) x} - \sqrt{x (x + 1)} + \sqrt{x + 1} \sqrt{x + 1})$

Raise $\sqrt{x + 1}$ to the power of $1$ .

$- (x - \sqrt{(x + 1) x} - \sqrt{x (x + 1)} + {\sqrt{x + 1}}^{1} \sqrt{x + 1})$

Raise $\sqrt{x + 1}$ to the power of $1$ .

$- (x - \sqrt{(x + 1) x} - \sqrt{x (x + 1)} + {\sqrt{x + 1}}^{1} {\sqrt{x + 1}}^{1})$

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

$- (x - \sqrt{(x + 1) x} - \sqrt{x (x + 1)} + {\sqrt{x + 1}}^{1 + 1})$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x + 1}}^{2}$ as $x + 1$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$- (x - \sqrt{(x + 1) x} - \sqrt{x (x + 1)} + {(x + 1)}^{1})$

Simplify.

$- (x - \sqrt{(x + 1) x} - \sqrt{x (x + 1)} + x + 1)$

Add $x$ and $x$ .

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

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

Reorder $x + 1$ and $x$ .

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

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

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

Apply the distributive property.

$- (2 x) - (- 2 \sqrt{x (x + 1)}) - 1 \cdot 1$

Simplify.

Multiply $2$ by $- 1$ .

$- 2 x - (- 2 \sqrt{x (x + 1)}) - 1 \cdot 1$

Multiply $- 2$ by $- 1$ .

$- 2 x + 2 \sqrt{x (x + 1)} - 1 \cdot 1$

Multiply $- 1$ by $1$ .

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

Simplify ( square root of x- square root of x+1)/( square root of x+ square root of x+1)

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