Simplify (6 square root of 8-6 square root of 2)(2 square root of 2+1)

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

Step 1

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$(6 \sqrt{4 (2)} - 6 \sqrt{2}) (2 \sqrt{2} + 1)$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $6$ .

$(12 \sqrt{2} - 6 \sqrt{2}) (2 \sqrt{2} + 1)$

Step 2

Simplify terms.

Subtract $6 \sqrt{2}$ from $12 \sqrt{2}$ .

$6 \sqrt{2} (2 \sqrt{2} + 1)$

Apply the distributive property.

$6 \sqrt{2} (2 \sqrt{2}) + 6 \sqrt{2} \cdot 1$

Step 3

Multiply $6 \sqrt{2} (2 \sqrt{2})$ .

Multiply $2$ by $6$ .

$12 \sqrt{2} \sqrt{2} + 6 \sqrt{2} \cdot 1$

Raise $\sqrt{2}$ to the power of $1$ .

$12 ({\sqrt{2}}^{1} \sqrt{2}) + 6 \sqrt{2} \cdot 1$

Raise $\sqrt{2}$ to the power of $1$ .

$12 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 6 \sqrt{2} \cdot 1$

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

$12 {\sqrt{2}}^{1 + 1} + 6 \sqrt{2} \cdot 1$

Add $1$ and $1$ .

$12 {\sqrt{2}}^{2} + 6 \sqrt{2} \cdot 1$

Step 4

Multiply $6$ by $1$ .

$12 {\sqrt{2}}^{2} + 6 \sqrt{2}$

Step 5

Simplify each term.

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

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

$12 {(2^{\frac{1}{2}})}^{2} + 6 \sqrt{2}$

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

$12 \cdot 2^{\frac{1}{2} \cdot 2} + 6 \sqrt{2}$

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

$12 \cdot 2^{\frac{2}{2}} + 6 \sqrt{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$12 \cdot 2^{\frac{2}{2}} + 6 \sqrt{2}$

Rewrite the expression.

$12 \cdot 2^{1} + 6 \sqrt{2}$

Evaluate the exponent.

$12 \cdot 2 + 6 \sqrt{2}$

Multiply $12$ by $2$ .

$24 + 6 \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$24 + 6 \sqrt{2}$

Decimal Form:

$32.48528137 \dots$

Simplify (6 square root of 8-6 square root of 2)(2 square root of 2+1)

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