Algebra Examples

$\frac{\sqrt{56} (\sqrt{28} + \sqrt{21} - \sqrt{7})}{\sqrt{2}}$

Step 1

Combine $\sqrt{56}$ and $\sqrt{2}$ into a single radical.

$\sqrt{\frac{56}{2}} (\sqrt{28} + \sqrt{21} - \sqrt{7})$

Step 2

Divide $56$ by $2$ .

$\sqrt{28} (\sqrt{28} + \sqrt{21} - \sqrt{7})$

Step 3

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 3.1

Factor $4$ out of $28$ .

$\sqrt{4 (7)} (\sqrt{28} + \sqrt{21} - \sqrt{7})$

Step 3.2

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

$\sqrt{2^{2} \cdot 7} (\sqrt{28} + \sqrt{21} - \sqrt{7})$

Step 4

Pull terms out from under the radical.

$2 \sqrt{7} (\sqrt{28} + \sqrt{21} - \sqrt{7})$

Step 5

Simplify each term.

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Step 5.1

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 5.1.1

Factor $4$ out of $28$ .

$2 \sqrt{7} (\sqrt{4 (7)} + \sqrt{21} - \sqrt{7})$

Step 5.1.2

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

$2 \sqrt{7} (\sqrt{2^{2} \cdot 7} + \sqrt{21} - \sqrt{7})$

Step 5.2

Pull terms out from under the radical.

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

Step 6

Simplify terms.

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Step 6.1

Subtract $\sqrt{7}$ from $2 \sqrt{7}$ .

$2 \sqrt{7} (\sqrt{7} + \sqrt{21})$

Step 6.2

Apply the distributive property.

$2 \sqrt{7} \sqrt{7} + 2 \sqrt{7} \sqrt{21}$

Step 7

Multiply $2 \sqrt{7} \sqrt{7}$ .

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Step 7.1

Raise $\sqrt{7}$ to the power of $1$ .

$2 ({\sqrt{7}}^{1} \sqrt{7}) + 2 \sqrt{7} \sqrt{21}$

Step 7.2

Raise $\sqrt{7}$ to the power of $1$ .

$2 ({\sqrt{7}}^{1} {\sqrt{7}}^{1}) + 2 \sqrt{7} \sqrt{21}$

Step 7.3

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

$2 {\sqrt{7}}^{1 + 1} + 2 \sqrt{7} \sqrt{21}$

Step 7.4

Add $1$ and $1$ .

$2 {\sqrt{7}}^{2} + 2 \sqrt{7} \sqrt{21}$

Step 8

Multiply $2 \sqrt{7} \sqrt{21}$ .

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Step 8.1

Combine using the product rule for radicals.

$2 {\sqrt{7}}^{2} + 2 \sqrt{21 \cdot 7}$

Step 8.2

Multiply $21$ by $7$ .

$2 {\sqrt{7}}^{2} + 2 \sqrt{147}$

Step 9

Simplify each term.

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Step 9.1

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 9.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$2 {(7^{\frac{1}{2}})}^{2} + 2 \sqrt{147}$

Step 9.1.2

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

$2 \cdot 7^{\frac{1}{2} \cdot 2} + 2 \sqrt{147}$

Step 9.1.3

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

$2 \cdot 7^{\frac{2}{2}} + 2 \sqrt{147}$

Step 9.1.4

Cancel the common factor of $2$ .

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Step 9.1.4.1

Cancel the common factor.

$2 \cdot 7^{\frac{2}{2}} + 2 \sqrt{147}$

Step 9.1.4.2

Rewrite the expression.

$2 \cdot 7^{1} + 2 \sqrt{147}$

Step 9.1.5

Evaluate the exponent.

$2 \cdot 7 + 2 \sqrt{147}$

Step 9.2

Multiply $2$ by $7$ .

$14 + 2 \sqrt{147}$

Step 9.3

Rewrite $147$ as $7^{2} \cdot 3$ .

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Step 9.3.1

Factor $49$ out of $147$ .

$14 + 2 \sqrt{49 (3)}$

Step 9.3.2

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

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

Step 9.4

Pull terms out from under the radical.

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

Step 9.5

Multiply $7$ by $2$ .

$14 + 14 \sqrt{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$14 + 14 \sqrt{3}$

Decimal Form:

$38.24871130 \dots$