Simplify the Radical Expression 2 square root of 3( square root of 27-8 square root of 6)

$2 \sqrt{3} (\sqrt{27} - 8 \sqrt{6})$

Step 1

Simplify each term.

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$2 \sqrt{3} (\sqrt{9 (3)} - 8 \sqrt{6})$

Rewrite $9$ as $3^{2}$ .

$2 \sqrt{3} (\sqrt{3^{2} \cdot 3} - 8 \sqrt{6})$

Pull terms out from under the radical.

$2 \sqrt{3} (| 3 | \sqrt{3} - 8 \sqrt{6})$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$2 \sqrt{3} (3 \sqrt{3} - 8 \sqrt{6})$

Step 2

Apply the distributive property.

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

Step 3

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

Multiply $3$ by $2$ .

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

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

$6 ({\sqrt{3}}^{1} \sqrt{3}) + 2 \sqrt{3} (- 8 \sqrt{6})$

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

$6 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + 2 \sqrt{3} (- 8 \sqrt{6})$

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

$6 {\sqrt{3}}^{1 + 1} + 2 \sqrt{3} (- 8 \sqrt{6})$

Add $1$ and $1$ .

$6 {\sqrt{3}}^{2} + 2 \sqrt{3} (- 8 \sqrt{6})$

Step 4

Multiply $2 \sqrt{3} (- 8 \sqrt{6})$ .

Multiply $- 8$ by $2$ .

$6 {\sqrt{3}}^{2} - 16 \sqrt{3} \sqrt{6}$

Combine using the product rule for radicals.

$6 {\sqrt{3}}^{2} - 16 \sqrt{6 \cdot 3}$

Multiply $6$ by $3$ .

$6 {\sqrt{3}}^{2} - 16 \sqrt{18}$

Step 5

Simplify each term.

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

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

$6 {(3^{\frac{1}{2}})}^{2} - 16 \sqrt{18}$

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

$6 \cdot 3^{\frac{1}{2} \cdot 2} - 16 \sqrt{18}$

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

$6 \cdot 3^{\frac{2}{2}} - 16 \sqrt{18}$

Cancel the common factor of $2$ .

Cancel the common factor.

$6 \cdot 3^{\frac{2}{2}} - 16 \sqrt{18}$

Rewrite the expression.

$6 \cdot 3^{1} - 16 \sqrt{18}$

Evaluate the exponent.

$6 \cdot 3 - 16 \sqrt{18}$

Multiply $6$ by $3$ .

$18 - 16 \sqrt{18}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$18 - 16 \sqrt{9 (2)}$

Rewrite $9$ as $3^{2}$ .

$18 - 16 \sqrt{3^{2} \cdot 2}$

Pull terms out from under the radical.

$18 - 16 (| 3 | \sqrt{2})$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$18 - 16 (3 \sqrt{2})$

Multiply $3$ by $- 16$ .

$18 - 48 \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$18 - 48 \sqrt{2}$

Decimal Form:

$- 49.88225099 \dots$

Simplify the Radical Expression 2 square root of 3( square root of 27-8 square root of 6)

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