Simplify ( square root of 1+x-x/(2 square root of 1+x))/(1+x)

$\frac{\sqrt{1 + x} - \frac{x}{2 \sqrt{1 + x}}}{1 + x}$

Multiply the numerator and denominator of the complex fraction by $2 \sqrt{1 + x}$ .

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

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

Combine.

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

Apply the distributive property.

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

Cancel the common factor of $2 \sqrt{1 + x}$ .

Move the leading negative in $- \frac{x}{2 \sqrt{1 + x}}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Multiply $2 \sqrt{1 + x} \sqrt{1 + x}$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $2$ by $1$ .

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

Subtract $x$ from $2 x$ .

$\frac{2 + x}{2 \sqrt{1 + x} \cdot 1 + 2 \sqrt{1 + x} x}$

Factor $2 \sqrt{1 + x}$ out of $2 \sqrt{1 + x} \cdot 1 + 2 \sqrt{1 + x} x$ .

Factor $2 \sqrt{1 + x}$ out of $2 \sqrt{1 + x} \cdot 1$ .

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

Factor $2 \sqrt{1 + x}$ out of $2 \sqrt{1 + x} x$ .

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

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

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

Multiply $\frac{2 + x}{2 \sqrt{1 + x} (1 + x)}$ by $\frac{\sqrt{1 + x}}{\sqrt{1 + x}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{2 + x}{2 \sqrt{1 + x} (1 + x)}$ by $\frac{\sqrt{1 + x}}{\sqrt{1 + x}}$ .

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

Move $\sqrt{1 + x}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the denominator.

Raise $1 + x$ to the power of $1$ .

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

Raise $1 + x$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Reorder factors in $\frac{(2 + x) \sqrt{1 + x}}{2 {(1 + x)}^{2}}$ .

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

Simplify ( square root of 1+x-x/(2 square root of 1+x))/(1+x)

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