Evaluate ( square root of 5- square root of 6)/( square root of 5+ square root of 6)

$\frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} + \sqrt{6}}$

Step 1

Multiply $\frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} + \sqrt{6}}$ by $\frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} - \sqrt{6}}$ .

$\frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} + \sqrt{6}} \cdot \frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} - \sqrt{6}}$

Step 2

Combine fractions.

Multiply $\frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} + \sqrt{6}}$ by $\frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} - \sqrt{6}}$ .

$\frac{(\sqrt{5} - \sqrt{6}) (\sqrt{5} - \sqrt{6})}{(\sqrt{5} + \sqrt{6}) (\sqrt{5} - \sqrt{6})}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{5} - \sqrt{6}) (\sqrt{5} - \sqrt{6})}{{\sqrt{5}}^{2} - \sqrt{30} + \sqrt{30} - {\sqrt{6}}^{2}}$

Simplify.

$\frac{(\sqrt{5} - \sqrt{6}) (\sqrt{5} - \sqrt{6})}{- 1}$

Simplify the expression.

Move the negative one from the denominator of $\frac{(\sqrt{5} - \sqrt{6}) (\sqrt{5} - \sqrt{6})}{- 1}$ .

$- 1 \cdot ((\sqrt{5} - \sqrt{6}) (\sqrt{5} - \sqrt{6}))$

Rewrite $- 1 \cdot ((\sqrt{5} - \sqrt{6}) (\sqrt{5} - \sqrt{6}))$ as $- ((\sqrt{5} - \sqrt{6}) (\sqrt{5} - \sqrt{6}))$ .

$- ((\sqrt{5} - \sqrt{6}) (\sqrt{5} - \sqrt{6}))$

Step 3

Expand $(\sqrt{5} - \sqrt{6}) (\sqrt{5} - \sqrt{6})$ using the FOIL Method.

Apply the distributive property.

$- (\sqrt{5} (\sqrt{5} - \sqrt{6}) - \sqrt{6} (\sqrt{5} - \sqrt{6}))$

Apply the distributive property.

$- (\sqrt{5} \sqrt{5} + \sqrt{5} (- \sqrt{6}) - \sqrt{6} (\sqrt{5} - \sqrt{6}))$

Apply the distributive property.

$- (\sqrt{5} \sqrt{5} + \sqrt{5} (- \sqrt{6}) - \sqrt{6} \sqrt{5} - \sqrt{6} (- \sqrt{6}))$

Step 4

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$- (\sqrt{5 \cdot 5} + \sqrt{5} (- \sqrt{6}) - \sqrt{6} \sqrt{5} - \sqrt{6} (- \sqrt{6}))$

Multiply $5$ by $5$ .

$- (\sqrt{25} + \sqrt{5} (- \sqrt{6}) - \sqrt{6} \sqrt{5} - \sqrt{6} (- \sqrt{6}))$

Rewrite $25$ as $5^{2}$ .

$- (\sqrt{5^{2}} + \sqrt{5} (- \sqrt{6}) - \sqrt{6} \sqrt{5} - \sqrt{6} (- \sqrt{6}))$

Pull terms out from under the radical, assuming positive real numbers.

$- (5 + \sqrt{5} (- \sqrt{6}) - \sqrt{6} \sqrt{5} - \sqrt{6} (- \sqrt{6}))$

Multiply $\sqrt{5} (- \sqrt{6})$ .

Combine using the product rule for radicals.

$- (5 - \sqrt{5 \cdot 6} - \sqrt{6} \sqrt{5} - \sqrt{6} (- \sqrt{6}))$

Multiply $5$ by $6$ .

$- (5 - \sqrt{30} - \sqrt{6} \sqrt{5} - \sqrt{6} (- \sqrt{6}))$

Multiply $- \sqrt{6} \sqrt{5}$ .

Combine using the product rule for radicals.

$- (5 - \sqrt{30} - \sqrt{5 \cdot 6} - \sqrt{6} (- \sqrt{6}))$

Multiply $5$ by $6$ .

$- (5 - \sqrt{30} - \sqrt{30} - \sqrt{6} (- \sqrt{6}))$

Multiply $- \sqrt{6} (- \sqrt{6})$ .

Multiply $- 1$ by $- 1$ .

$- (5 - \sqrt{30} - \sqrt{30} + 1 \sqrt{6} \sqrt{6})$

Multiply $\sqrt{6}$ by $1$ .

$- (5 - \sqrt{30} - \sqrt{30} + \sqrt{6} \sqrt{6})$

Raise $\sqrt{6}$ to the power of $1$ .

$- (5 - \sqrt{30} - \sqrt{30} + {\sqrt{6}}^{1} \sqrt{6})$

Raise $\sqrt{6}$ to the power of $1$ .

$- (5 - \sqrt{30} - \sqrt{30} + {\sqrt{6}}^{1} {\sqrt{6}}^{1})$

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

$- (5 - \sqrt{30} - \sqrt{30} + {\sqrt{6}}^{1 + 1})$

Add $1$ and $1$ .

$- (5 - \sqrt{30} - \sqrt{30} + {\sqrt{6}}^{2})$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$- (5 - \sqrt{30} - \sqrt{30} + {(6^{\frac{1}{2}})}^{2})$

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

$- (5 - \sqrt{30} - \sqrt{30} + 6^{\frac{1}{2} \cdot 2})$

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

$- (5 - \sqrt{30} - \sqrt{30} + 6^{\frac{2}{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$- (5 - \sqrt{30} - \sqrt{30} + 6^{\frac{2}{2}})$

Rewrite the expression.

$- (5 - \sqrt{30} - \sqrt{30} + 6^{1})$

Evaluate the exponent.

$- (5 - \sqrt{30} - \sqrt{30} + 6)$

Add $5$ and $6$ .

$- (11 - \sqrt{30} - \sqrt{30})$

Subtract $\sqrt{30}$ from $- \sqrt{30}$ .

$- (11 - 2 \sqrt{30})$

Step 5

Apply the distributive property.

$- 1 \cdot 11 - (- 2 \sqrt{30})$

Step 6

Multiply.

Multiply $- 1$ by $11$ .

$- 11 - (- 2 \sqrt{30})$

Multiply $- 2$ by $- 1$ .

$- 11 + 2 \sqrt{30}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 11 + 2 \sqrt{30}$

Decimal Form:

$- 0.04554884 \dots$

Evaluate ( square root of 5- square root of 6)/( square root of 5+ square root of 6)

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