Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1.1

Factor $4$ out of $8$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine fractions.

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Step 3.1

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

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

Step 3.2

Expand the denominator using the FOIL method.

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

Step 3.3

Simplify.

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

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Step 5.1.1.1

Combine using the product rule for radicals.

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

Step 5.1.1.2

Multiply $10$ by $2$ .

$\frac{2 \sqrt{20} + 2 \sqrt{2} \sqrt{2} - \sqrt{10} \sqrt{10} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.2

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

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Step 5.1.2.1

Factor $4$ out of $20$ .

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

Step 5.1.2.2

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

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

Step 5.1.3

Pull terms out from under the radical.

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

Step 5.1.4

Multiply $2$ by $2$ .

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

Step 5.1.5

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

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Step 5.1.5.1

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

$\frac{4 \sqrt{5} + 2 ({\sqrt{2}}^{1} \sqrt{2}) - \sqrt{10} \sqrt{10} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.5.2

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

$\frac{4 \sqrt{5} + 2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) - \sqrt{10} \sqrt{10} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.5.3

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

$\frac{4 \sqrt{5} + 2 {\sqrt{2}}^{1 + 1} - \sqrt{10} \sqrt{10} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.5.4

Add $1$ and $1$ .

$\frac{4 \sqrt{5} + 2 {\sqrt{2}}^{2} - \sqrt{10} \sqrt{10} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.6

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

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Step 5.1.6.1

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

$\frac{4 \sqrt{5} + 2 {(2^{\frac{1}{2}})}^{2} - \sqrt{10} \sqrt{10} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.6.2

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

$\frac{4 \sqrt{5} + 2 \cdot 2^{\frac{1}{2} \cdot 2} - \sqrt{10} \sqrt{10} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.6.3

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

$\frac{4 \sqrt{5} + 2 \cdot 2^{\frac{2}{2}} - \sqrt{10} \sqrt{10} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.6.4

Cancel the common factor of $2$ .

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Step 5.1.6.4.1

Cancel the common factor.

$\frac{4 \sqrt{5} + 2 \cdot 2^{\frac{2}{2}} - \sqrt{10} \sqrt{10} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.6.4.2

Rewrite the expression.

$\frac{4 \sqrt{5} + 2 \cdot 2^{1} - \sqrt{10} \sqrt{10} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.6.5

Evaluate the exponent.

$\frac{4 \sqrt{5} + 2 \cdot 2 - \sqrt{10} \sqrt{10} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.7

Multiply $2$ by $2$ .

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

Step 5.1.8

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

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Step 5.1.8.1

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

$\frac{4 \sqrt{5} + 4 - ({\sqrt{10}}^{1} \sqrt{10}) - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.8.2

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

$\frac{4 \sqrt{5} + 4 - ({\sqrt{10}}^{1} {\sqrt{10}}^{1}) - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.8.3

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

$\frac{4 \sqrt{5} + 4 - {\sqrt{10}}^{1 + 1} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.8.4

Add $1$ and $1$ .

$\frac{4 \sqrt{5} + 4 - {\sqrt{10}}^{2} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.9

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

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Step 5.1.9.1

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

$\frac{4 \sqrt{5} + 4 - {(10^{\frac{1}{2}})}^{2} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.9.2

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

$\frac{4 \sqrt{5} + 4 - 10^{\frac{1}{2} \cdot 2} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.9.3

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

$\frac{4 \sqrt{5} + 4 - 10^{\frac{2}{2}} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.9.4

Cancel the common factor of $2$ .

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Step 5.1.9.4.1

Cancel the common factor.

$\frac{4 \sqrt{5} + 4 - 10^{\frac{2}{2}} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.9.4.2

Rewrite the expression.

$\frac{4 \sqrt{5} + 4 - 10^{1} - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.9.5

Evaluate the exponent.

$\frac{4 \sqrt{5} + 4 - 1 \cdot 10 - \sqrt{10} \sqrt{2}}{8}$

Step 5.1.10

Multiply $- 1$ by $10$ .

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

Step 5.1.11

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

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Step 5.1.11.1

Combine using the product rule for radicals.

$\frac{4 \sqrt{5} + 4 - 10 - \sqrt{2 \cdot 10}}{8}$

Step 5.1.11.2

Multiply $2$ by $10$ .

$\frac{4 \sqrt{5} + 4 - 10 - \sqrt{20}}{8}$

Step 5.1.12

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

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Step 5.1.12.1

Factor $4$ out of $20$ .

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

Step 5.1.12.2

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

$\frac{4 \sqrt{5} + 4 - 10 - \sqrt{2^{2} \cdot 5}}{8}$

Step 5.1.13

Pull terms out from under the radical.

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

Step 5.1.14

Multiply $2$ by $- 1$ .

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

Step 5.2

Subtract $2 \sqrt{5}$ from $4 \sqrt{5}$ .

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

Step 5.3

Subtract $10$ from $4$ .

$\frac{- 6 + 2 \sqrt{5}}{8}$

Step 6

Cancel the common factor of $- 6 + 2 \sqrt{5}$ and $8$ .

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Step 6.1

Factor $2$ out of $- 6$ .

$\frac{2 \cdot - 3 + 2 \sqrt{5}}{8}$

Step 6.2

Factor $2$ out of $2 \sqrt{5}$ .

$\frac{2 \cdot - 3 + 2 (\sqrt{5})}{8}$

Step 6.3

Factor $2$ out of $2 \cdot - 3 + 2 (\sqrt{5})$ .

$\frac{2 \cdot (- 3 + \sqrt{5})}{8}$

Step 6.4

Cancel the common factors.

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Step 6.4.1

Factor $2$ out of $8$ .

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

Step 6.4.2

Cancel the common factor.

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

Step 6.4.3

Rewrite the expression.

$\frac{- 3 + \sqrt{5}}{4}$

Step 7

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{- 1 (3) + \sqrt{5}}{4}$

Step 8

Factor $- 1$ out of $\sqrt{5}$ .

$\frac{- 1 (3) - 1 (- \sqrt{5})}{4}$

Step 9

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{5})$ .

$\frac{- 1 (3 - \sqrt{5})}{4}$

Step 10

Move the negative in front of the fraction.

$- \frac{3 - \sqrt{5}}{4}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{3 - \sqrt{5}}{4}$

Decimal Form:

$- 0.19098300 \dots$