Algebra Examples

$\frac{\sqrt{40} - 2 \sqrt{5}}{\sqrt{10}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 1.1.1

Factor $4$ out of $40$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{2 \sqrt{10} - 2 \sqrt{5}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{2 \sqrt{10} - 2 \sqrt{5}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{2 \sqrt{10} - 2 \sqrt{5}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Evaluate the exponent.

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

Step 4

Cancel the common factor of $2 \sqrt{10} - 2 \sqrt{5}$ and $10$ .

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Step 4.1

Factor $2$ out of $(2 \sqrt{10} - 2 \sqrt{5}) \sqrt{10}$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2$ out of $10$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Apply the distributive property.

$\frac{\sqrt{10} \sqrt{10} - \sqrt{5} \sqrt{10}}{5}$

Step 6

Combine using the product rule for radicals.

$\frac{\sqrt{10 \cdot 10} - \sqrt{5} \sqrt{10}}{5}$

Step 7

Multiply $- \sqrt{5} \sqrt{10}$ .

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Step 7.1

Combine using the product rule for radicals.

$\frac{\sqrt{10 \cdot 10} - \sqrt{10 \cdot 5}}{5}$

Step 7.2

Multiply $10$ by $5$ .

$\frac{\sqrt{10 \cdot 10} - \sqrt{50}}{5}$

Step 8

Simplify each term.

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Step 8.1

Multiply $10$ by $10$ .

$\frac{\sqrt{100} - \sqrt{50}}{5}$

Step 8.2

Rewrite $100$ as $10^{2}$ .

$\frac{\sqrt{10^{2}} - \sqrt{50}}{5}$

Step 8.3

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

$\frac{10 - \sqrt{50}}{5}$

Step 8.4

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 8.4.1

Factor $25$ out of $50$ .

$\frac{10 - \sqrt{25 (2)}}{5}$

Step 8.4.2

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

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

Step 8.5

Pull terms out from under the radical.

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

Step 8.6

Multiply $5$ by $- 1$ .

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

Step 9

Cancel the common factor of $10 - 5 \sqrt{2}$ and $5$ .

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Step 9.1

Factor $5$ out of $10$ .

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $5$ out of $5$ .

$\frac{5 (2 - \sqrt{2})}{5 (1)}$

Step 9.4.2

Cancel the common factor.

$\frac{5 (2 - \sqrt{2})}{5 \cdot 1}$

Step 9.4.3

Rewrite the expression.

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

Step 9.4.4

Divide $2 - \sqrt{2}$ by $1$ .

$2 - \sqrt{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$2 - \sqrt{2}$

Decimal Form:

$0.58578643 \dots$