Algebra Examples

$\sqrt{42} \cdot (\sqrt{6} + \sqrt{7} + \sqrt{14} - \sqrt{21})$

Step 1

Apply the distributive property.

$\sqrt{42} \sqrt{6} + \sqrt{42} \sqrt{7} + \sqrt{42} \sqrt{14} + \sqrt{42} (- \sqrt{21})$

Step 2

Simplify.

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Step 2.1

Combine using the product rule for radicals.

$\sqrt{42 \cdot 6} + \sqrt{42} \sqrt{7} + \sqrt{42} \sqrt{14} + \sqrt{42} (- \sqrt{21})$

Step 2.2

Combine using the product rule for radicals.

$\sqrt{42 \cdot 6} + \sqrt{42 \cdot 7} + \sqrt{42} \sqrt{14} + \sqrt{42} (- \sqrt{21})$

Step 2.3

Combine using the product rule for radicals.

$\sqrt{42 \cdot 6} + \sqrt{42 \cdot 7} + \sqrt{42 \cdot 14} + \sqrt{42} (- \sqrt{21})$

Step 2.4

Multiply $\sqrt{42} (- \sqrt{21})$ .

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Step 2.4.1

Combine using the product rule for radicals.

$\sqrt{42 \cdot 6} + \sqrt{42 \cdot 7} + \sqrt{42 \cdot 14} - \sqrt{42 \cdot 21}$

Step 2.4.2

Multiply $42$ by $21$ .

$\sqrt{42 \cdot 6} + \sqrt{42 \cdot 7} + \sqrt{42 \cdot 14} - \sqrt{882}$

Step 3

Simplify each term.

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Step 3.1

Multiply $42$ by $6$ .

$\sqrt{252} + \sqrt{42 \cdot 7} + \sqrt{42 \cdot 14} - \sqrt{882}$

Step 3.2

Rewrite $252$ as $6^{2} \cdot 7$ .

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Step 3.2.1

Factor $36$ out of $252$ .

$\sqrt{36 (7)} + \sqrt{42 \cdot 7} + \sqrt{42 \cdot 14} - \sqrt{882}$

Step 3.2.2

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2} \cdot 7} + \sqrt{42 \cdot 7} + \sqrt{42 \cdot 14} - \sqrt{882}$

Step 3.3

Pull terms out from under the radical.

$6 \sqrt{7} + \sqrt{42 \cdot 7} + \sqrt{42 \cdot 14} - \sqrt{882}$

Step 3.4

Multiply $42$ by $7$ .

$6 \sqrt{7} + \sqrt{294} + \sqrt{42 \cdot 14} - \sqrt{882}$

Step 3.5

Rewrite $294$ as $7^{2} \cdot 6$ .

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Step 3.5.1

Factor $49$ out of $294$ .

$6 \sqrt{7} + \sqrt{49 (6)} + \sqrt{42 \cdot 14} - \sqrt{882}$

Step 3.5.2

Rewrite $49$ as $7^{2}$ .

$6 \sqrt{7} + \sqrt{7^{2} \cdot 6} + \sqrt{42 \cdot 14} - \sqrt{882}$

Step 3.6

Pull terms out from under the radical.

$6 \sqrt{7} + 7 \sqrt{6} + \sqrt{42 \cdot 14} - \sqrt{882}$

Step 3.7

Multiply $42$ by $14$ .

$6 \sqrt{7} + 7 \sqrt{6} + \sqrt{588} - \sqrt{882}$

Step 3.8

Rewrite $588$ as $14^{2} \cdot 3$ .

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Step 3.8.1

Factor $196$ out of $588$ .

$6 \sqrt{7} + 7 \sqrt{6} + \sqrt{196 (3)} - \sqrt{882}$

Step 3.8.2

Rewrite $196$ as $14^{2}$ .

$6 \sqrt{7} + 7 \sqrt{6} + \sqrt{14^{2} \cdot 3} - \sqrt{882}$

Step 3.9

Pull terms out from under the radical.

$6 \sqrt{7} + 7 \sqrt{6} + 14 \sqrt{3} - \sqrt{882}$

Step 3.10

Rewrite $882$ as $21^{2} \cdot 2$ .

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Step 3.10.1

Factor $441$ out of $882$ .

$6 \sqrt{7} + 7 \sqrt{6} + 14 \sqrt{3} - \sqrt{441 (2)}$

Step 3.10.2

Rewrite $441$ as $21^{2}$ .

$6 \sqrt{7} + 7 \sqrt{6} + 14 \sqrt{3} - \sqrt{21^{2} \cdot 2}$

Step 3.11

Pull terms out from under the radical.

$6 \sqrt{7} + 7 \sqrt{6} + 14 \sqrt{3} - (21 \sqrt{2})$

Step 3.12

Multiply $21$ by $- 1$ .

$6 \sqrt{7} + 7 \sqrt{6} + 14 \sqrt{3} - 21 \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt{7} + 7 \sqrt{6} + 14 \sqrt{3} - 21 \sqrt{2}$

Decimal Form:

$27.57116256 \dots$