Algebra Examples

$2 \sqrt{48} - (2 + \sqrt{12}) (5 - \sqrt{12})$

Step 1

Simplify each term.

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Step 1.1

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 1.1.1

Factor $16$ out of $48$ .

$2 \sqrt{16 (3)} - (2 + \sqrt{12}) (5 - \sqrt{12})$

Step 1.1.2

Rewrite $16$ as $4^{2}$ .

$2 \sqrt{4^{2} \cdot 3} - (2 + \sqrt{12}) (5 - \sqrt{12})$

Step 1.2

Pull terms out from under the radical.

$2 (4 \sqrt{3}) - (2 + \sqrt{12}) (5 - \sqrt{12})$

Step 1.3

Multiply $4$ by $2$ .

$8 \sqrt{3} - (2 + \sqrt{12}) (5 - \sqrt{12})$

Step 1.4

Simplify each term.

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Step 1.4.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.4.1.1

Factor $4$ out of $12$ .

$8 \sqrt{3} - (2 + \sqrt{4 (3)}) (5 - \sqrt{12})$

Step 1.4.1.2

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

$8 \sqrt{3} - (2 + \sqrt{2^{2} \cdot 3}) (5 - \sqrt{12})$

Step 1.4.2

Pull terms out from under the radical.

$8 \sqrt{3} - (2 + 2 \sqrt{3}) (5 - \sqrt{12})$

Step 1.5

Apply the distributive property.

$8 \sqrt{3} + (- 1 \cdot 2 - (2 \sqrt{3})) (5 - \sqrt{12})$

Step 1.6

Multiply $- 1$ by $2$ .

$8 \sqrt{3} + (- 2 - (2 \sqrt{3})) (5 - \sqrt{12})$

Step 1.7

Multiply $2$ by $- 1$ .

$8 \sqrt{3} + (- 2 - 2 \sqrt{3}) (5 - \sqrt{12})$

Step 1.8

Simplify each term.

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Step 1.8.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.8.1.1

Factor $4$ out of $12$ .

$8 \sqrt{3} + (- 2 - 2 \sqrt{3}) (5 - \sqrt{4 (3)})$

Step 1.8.1.2

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

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

Step 1.8.2

Pull terms out from under the radical.

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

Step 1.8.3

Multiply $2$ by $- 1$ .

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

Step 1.9

Expand $(- 2 - 2 \sqrt{3}) (5 - 2 \sqrt{3})$ using the FOIL Method.

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Step 1.9.1

Apply the distributive property.

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

Step 1.9.2

Apply the distributive property.

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

Step 1.9.3

Apply the distributive property.

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

Step 1.10

Simplify and combine like terms.

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Step 1.10.1

Simplify each term.

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Step 1.10.1.1

Multiply $- 2$ by $5$ .

$8 \sqrt{3} - 10 - 2 (- 2 \sqrt{3}) - 2 \sqrt{3} \cdot 5 - 2 \sqrt{3} (- 2 \sqrt{3})$

Step 1.10.1.2

Multiply $- 2$ by $- 2$ .

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 2 \sqrt{3} \cdot 5 - 2 \sqrt{3} (- 2 \sqrt{3})$

Step 1.10.1.3

Multiply $5$ by $- 2$ .

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} - 2 \sqrt{3} (- 2 \sqrt{3})$

Step 1.10.1.4

Multiply $- 2 \sqrt{3} (- 2 \sqrt{3})$ .

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Step 1.10.1.4.1

Multiply $- 2$ by $- 2$ .

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 4 \sqrt{3} \sqrt{3}$

Step 1.10.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 4 ({\sqrt{3}}^{1} \sqrt{3})$

Step 1.10.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 1.10.1.4.4

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

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 4 {\sqrt{3}}^{1 + 1}$

Step 1.10.1.4.5

Add $1$ and $1$ .

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 4 {\sqrt{3}}^{2}$

Step 1.10.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.10.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 4 {(3^{\frac{1}{2}})}^{2}$

Step 1.10.1.5.2

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

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 4 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 1.10.1.5.3

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

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}$

Step 1.10.1.5.4

Cancel the common factor of $2$ .

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Step 1.10.1.5.4.1

Cancel the common factor.

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}$

Step 1.10.1.5.4.2

Rewrite the expression.

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 4 \cdot 3^{1}$

Step 1.10.1.5.5

Evaluate the exponent.

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 4 \cdot 3$

Step 1.10.1.6

Multiply $4$ by $3$ .

$8 \sqrt{3} - 10 + 4 \sqrt{3} - 10 \sqrt{3} + 12$

Step 1.10.2

Add $- 10$ and $12$ .

$8 \sqrt{3} + 2 + 4 \sqrt{3} - 10 \sqrt{3}$

Step 1.10.3

Subtract $10 \sqrt{3}$ from $4 \sqrt{3}$ .

$8 \sqrt{3} + 2 - 6 \sqrt{3}$

Step 2

Subtract $6 \sqrt{3}$ from $8 \sqrt{3}$ .

$2 + 2 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$2 + 2 \sqrt{3}$

Decimal Form:

$5.46410161 \dots$