Simplify the Radical Expression ( square root of 2- square root of 10)/( square root of 2+ square root of 10)

$\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}}$

Step 1

Multiply $\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}}$ by $\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} - \sqrt{10}}$ .

$\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}} \cdot \frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} - \sqrt{10}}$

Step 2

Multiply $\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} + \sqrt{10}}$ by $\frac{\sqrt{2} - \sqrt{10}}{\sqrt{2} - \sqrt{10}}$ .

$\frac{(\sqrt{2} - \sqrt{10}) (\sqrt{2} - \sqrt{10})}{(\sqrt{2} + \sqrt{10}) (\sqrt{2} - \sqrt{10})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{(\sqrt{2} - \sqrt{10}) (\sqrt{2} - \sqrt{10})}{{\sqrt{2}}^{2} - \sqrt{20} + \sqrt{20} - {\sqrt{10}}^{2}}$

Step 4

Simplify.

$\frac{(\sqrt{2} - \sqrt{10}) (\sqrt{2} - \sqrt{10})}{- 8}$

Step 5

Simplify the numerator.

Raise $\sqrt{2} - \sqrt{10}$ to the power of $1$ .

$\frac{{(\sqrt{2} - \sqrt{10})}^{1} (\sqrt{2} - \sqrt{10})}{- 8}$

Raise $\sqrt{2} - \sqrt{10}$ to the power of $1$ .

$\frac{{(\sqrt{2} - \sqrt{10})}^{1} {(\sqrt{2} - \sqrt{10})}^{1}}{- 8}$

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

$\frac{{(\sqrt{2} - \sqrt{10})}^{1 + 1}}{- 8}$

Add $1$ and $1$ .

$\frac{{(\sqrt{2} - \sqrt{10})}^{2}}{- 8}$

Step 6

Rewrite ${(\sqrt{2} - \sqrt{10})}^{2}$ as $(\sqrt{2} - \sqrt{10}) (\sqrt{2} - \sqrt{10})$ .

$\frac{(\sqrt{2} - \sqrt{10}) (\sqrt{2} - \sqrt{10})}{- 8}$

Step 7

Expand $(\sqrt{2} - \sqrt{10}) (\sqrt{2} - \sqrt{10})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{2} (\sqrt{2} - \sqrt{10}) - \sqrt{10} (\sqrt{2} - \sqrt{10})}{- 8}$

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{2} + \sqrt{2} (- \sqrt{10}) - \sqrt{10} (\sqrt{2} - \sqrt{10})}{- 8}$

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{2} + \sqrt{2} (- \sqrt{10}) - \sqrt{10} \sqrt{2} - \sqrt{10} (- \sqrt{10})}{- 8}$

Step 8

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 2} + \sqrt{2} (- \sqrt{10}) - \sqrt{10} \sqrt{2} - \sqrt{10} (- \sqrt{10})}{- 8}$

Multiply $2$ by $2$ .

$\frac{\sqrt{4} + \sqrt{2} (- \sqrt{10}) - \sqrt{10} \sqrt{2} - \sqrt{10} (- \sqrt{10})}{- 8}$

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2}} + \sqrt{2} (- \sqrt{10}) - \sqrt{10} \sqrt{2} - \sqrt{10} (- \sqrt{10})}{- 8}$

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

$\frac{2 + \sqrt{2} (- \sqrt{10}) - \sqrt{10} \sqrt{2} - \sqrt{10} (- \sqrt{10})}{- 8}$

Multiply $\sqrt{2} (- \sqrt{10})$ .

Combine using the product rule for radicals.

$\frac{2 - \sqrt{2 \cdot 10} - \sqrt{10} \sqrt{2} - \sqrt{10} (- \sqrt{10})}{- 8}$

Multiply $2$ by $10$ .

$\frac{2 - \sqrt{20} - \sqrt{10} \sqrt{2} - \sqrt{10} (- \sqrt{10})}{- 8}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$\frac{2 - \sqrt{4 (5)} - \sqrt{10} \sqrt{2} - \sqrt{10} (- \sqrt{10})}{- 8}$

Rewrite $4$ as $2^{2}$ .

$\frac{2 - \sqrt{2^{2} \cdot 5} - \sqrt{10} \sqrt{2} - \sqrt{10} (- \sqrt{10})}{- 8}$

Pull terms out from under the radical.

$\frac{2 - (2 \sqrt{5}) - \sqrt{10} \sqrt{2} - \sqrt{10} (- \sqrt{10})}{- 8}$

Multiply $2$ by $- 1$ .

$\frac{2 - 2 \sqrt{5} - \sqrt{10} \sqrt{2} - \sqrt{10} (- \sqrt{10})}{- 8}$

Multiply $- \sqrt{10} \sqrt{2}$ .

Combine using the product rule for radicals.

$\frac{2 - 2 \sqrt{5} - \sqrt{2 \cdot 10} - \sqrt{10} (- \sqrt{10})}{- 8}$

Multiply $2$ by $10$ .

$\frac{2 - 2 \sqrt{5} - \sqrt{20} - \sqrt{10} (- \sqrt{10})}{- 8}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$\frac{2 - 2 \sqrt{5} - \sqrt{4 (5)} - \sqrt{10} (- \sqrt{10})}{- 8}$

Rewrite $4$ as $2^{2}$ .

$\frac{2 - 2 \sqrt{5} - \sqrt{2^{2} \cdot 5} - \sqrt{10} (- \sqrt{10})}{- 8}$

Pull terms out from under the radical.

$\frac{2 - 2 \sqrt{5} - (2 \sqrt{5}) - \sqrt{10} (- \sqrt{10})}{- 8}$

Multiply $2$ by $- 1$ .

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} - \sqrt{10} (- \sqrt{10})}{- 8}$

Multiply $- \sqrt{10} (- \sqrt{10})$ .

Multiply $- 1$ by $- 1$ .

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + 1 \sqrt{10} \sqrt{10}}{- 8}$

Multiply $\sqrt{10}$ by $1$ .

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + \sqrt{10} \sqrt{10}}{- 8}$

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + {\sqrt{10}}^{1} \sqrt{10}}{- 8}$

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + {\sqrt{10}}^{1} {\sqrt{10}}^{1}}{- 8}$

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

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + {\sqrt{10}}^{1 + 1}}{- 8}$

Add $1$ and $1$ .

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + {\sqrt{10}}^{2}}{- 8}$

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + {(10^{\frac{1}{2}})}^{2}}{- 8}$

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

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + 10^{\frac{1}{2} \cdot 2}}{- 8}$

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

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + 10^{\frac{2}{2}}}{- 8}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + 10^{\frac{2}{2}}}{- 8}$

Rewrite the expression.

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + 10^{1}}{- 8}$

Evaluate the exponent.

$\frac{2 - 2 \sqrt{5} - 2 \sqrt{5} + 10}{- 8}$

Add $2$ and $10$ .

$\frac{12 - 2 \sqrt{5} - 2 \sqrt{5}}{- 8}$

Subtract $2 \sqrt{5}$ from $- 2 \sqrt{5}$ .

$\frac{12 - 4 \sqrt{5}}{- 8}$

Step 9

Cancel the common factor of $12 - 4 \sqrt{5}$ and $- 8$ .

Factor $4$ out of $12$ .

$\frac{4 (3) - 4 \sqrt{5}}{- 8}$

Factor $4$ out of $- 4 \sqrt{5}$ .

$\frac{4 (3) + 4 (- \sqrt{5})}{- 8}$

Factor $4$ out of $4 (3) + 4 (- \sqrt{5})$ .

$\frac{4 (3 - \sqrt{5})}{- 8}$

Cancel the common factors.

Factor $4$ out of $- 8$ .

$\frac{4 (3 - \sqrt{5})}{4 \cdot - 2}$

Cancel the common factor.

$\frac{4 (3 - \sqrt{5})}{4 \cdot - 2}$

Rewrite the expression.

$\frac{3 - \sqrt{5}}{- 2}$

Step 10

Move the negative in front of the fraction.

$- \frac{3 - \sqrt{5}}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{3 - \sqrt{5}}{2}$

Decimal Form:

$- 0.38196601 \dots$

Simplify the Radical Expression ( square root of 2- square root of 10)/( square root of 2+ square root of 10)

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