Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 2.4

Simplify the expression.

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

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

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

Step 3

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

Tap for more steps...

Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

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

Tap for more steps...

Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.1.4

Add $1$ and $1$ .

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

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.2.4.1

Cancel the common factor.

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

Step 4.1.2.4.2

Rewrite the expression.

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

Step 4.1.2.5

Simplify.

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

Step 4.1.3

Rewrite using the commutative property of multiplication.

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

Step 4.1.4

Combine using the product rule for radicals.

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

Step 4.1.5

Combine using the product rule for radicals.

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

Step 4.1.6

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

Tap for more steps...

Step 4.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.6.2

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

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

Step 4.1.6.3

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

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

Step 4.1.6.4

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

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

Step 4.1.6.5

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

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

Step 4.1.6.6

Add $1$ and $1$ .

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

Step 4.1.7

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

Tap for more steps...

Step 4.1.7.1

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

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

Step 4.1.7.2

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

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

Step 4.1.7.3

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

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

Step 4.1.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.7.4.1

Cancel the common factor.

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

Step 4.1.7.4.2

Rewrite the expression.

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

Step 4.1.7.5

Simplify.

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

Step 4.2

Add $x$ and $x$ .

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

Step 4.3

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

Tap for more steps...

Step 4.3.1

Reorder $x + 1$ and $x$ .

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

Step 4.3.2

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

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

Step 5

Apply the distributive property.

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

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply $2$ by $- 1$ .

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

Step 6.2

Multiply $- 2$ by $- 1$ .

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

Step 6.3

Multiply $- 1$ by $1$ .

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