Simplify 2 square root of 48-(2+ square root of 12)(5- square root of 12)

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

Step 1

Simplify each term.

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

Factor $16$ out of $48$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $4$ by $2$ .

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

Simplify each term.

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $- 1$ .

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

Simplify each term.

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 2$ by $5$ .

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

Multiply $- 2$ by $- 2$ .

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

Multiply $5$ by $- 2$ .

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

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

Multiply $- 2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $4$ by $3$ .

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

Add $- 10$ and $12$ .

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

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:

$5.46410161 \dots$

Simplify 2 square root of 48-(2+ square root of 12)(5- square root of 12)

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