Algebra Examples

$\sqrt{5} (\sqrt{20} - \sqrt{30}) + \sqrt{6}$

Step 1

Simplify each term.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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Step 1.1.1.1

Factor $4$ out of $20$ .

$\sqrt{5} (\sqrt{4 (5)} - \sqrt{30}) + \sqrt{6}$

Step 1.1.1.2

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

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

Step 1.1.2

Pull terms out from under the radical.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Multiply $\sqrt{5} (2 \sqrt{5})$ .

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Step 1.3.1

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 1.3.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 1.3.3

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

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

Step 1.3.4

Add $1$ and $1$ .

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

Step 1.4

Multiply $\sqrt{5} (- \sqrt{30})$ .

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Step 1.4.1

Combine using the product rule for radicals.

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

Step 1.4.2

Multiply $5$ by $30$ .

$2 {\sqrt{5}}^{2} - \sqrt{150} + \sqrt{6}$

Step 1.5

Simplify each term.

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Step 1.5.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$2 {(5^{\frac{1}{2}})}^{2} - \sqrt{150} + \sqrt{6}$

Step 1.5.1.2

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

$2 \cdot 5^{\frac{1}{2} \cdot 2} - \sqrt{150} + \sqrt{6}$

Step 1.5.1.3

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

$2 \cdot 5^{\frac{2}{2}} - \sqrt{150} + \sqrt{6}$

Step 1.5.1.4

Cancel the common factor of $2$ .

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Step 1.5.1.4.1

Cancel the common factor.

$2 \cdot 5^{\frac{2}{2}} - \sqrt{150} + \sqrt{6}$

Step 1.5.1.4.2

Rewrite the expression.

$2 \cdot 5^{1} - \sqrt{150} + \sqrt{6}$

Step 1.5.1.5

Evaluate the exponent.

$2 \cdot 5 - \sqrt{150} + \sqrt{6}$

Step 1.5.2

Multiply $2$ by $5$ .

$10 - \sqrt{150} + \sqrt{6}$

Step 1.5.3

Rewrite $150$ as $5^{2} \cdot 6$ .

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Step 1.5.3.1

Factor $25$ out of $150$ .

$10 - \sqrt{25 (6)} + \sqrt{6}$

Step 1.5.3.2

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

$10 - \sqrt{5^{2} \cdot 6} + \sqrt{6}$

Step 1.5.4

Pull terms out from under the radical.

$10 - (5 \sqrt{6}) + \sqrt{6}$

Step 1.5.5

Multiply $5$ by $- 1$ .

$10 - 5 \sqrt{6} + \sqrt{6}$

Step 2

Add $- 5 \sqrt{6}$ and $\sqrt{6}$ .

$10 - 4 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$10 - 4 \sqrt{6}$

Decimal Form:

$0.20204102 \dots$