Simplify 9 square root of 2( square root of 9+2 square root of 2)

$9 \sqrt{2} (\sqrt{9} + 2 \sqrt{2})$

Step 1

Simplify each term.

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

$9 \sqrt{2} (\sqrt{3^{2}} + 2 \sqrt{2})$

Pull terms out from under the radical, assuming positive real numbers.

$9 \sqrt{2} (3 + 2 \sqrt{2})$

Step 2

Simplify by multiplying through.

Apply the distributive property.

$9 \sqrt{2} \cdot 3 + 9 \sqrt{2} (2 \sqrt{2})$

Multiply $3$ by $9$ .

$27 \sqrt{2} + 9 \sqrt{2} (2 \sqrt{2})$

Step 3

Multiply $9 \sqrt{2} (2 \sqrt{2})$ .

Multiply $2$ by $9$ .

$27 \sqrt{2} + 18 \sqrt{2} \sqrt{2}$

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

$27 \sqrt{2} + 18 ({\sqrt{2}}^{1} \sqrt{2})$

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

$27 \sqrt{2} + 18 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

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

$27 \sqrt{2} + 18 {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$27 \sqrt{2} + 18 {\sqrt{2}}^{2}$

Step 4

Simplify each term.

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

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

$27 \sqrt{2} + 18 {(2^{\frac{1}{2}})}^{2}$

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

$27 \sqrt{2} + 18 \cdot 2^{\frac{1}{2} \cdot 2}$

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

$27 \sqrt{2} + 18 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$27 \sqrt{2} + 18 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$27 \sqrt{2} + 18 \cdot 2^{1}$

Evaluate the exponent.

$27 \sqrt{2} + 18 \cdot 2$

Multiply $18$ by $2$ .

$27 \sqrt{2} + 36$

Step 5

The result can be shown in multiple forms.

Exact Form:

$27 \sqrt{2} + 36$

Decimal Form:

$74.18376618 \dots$

Simplify 9 square root of 2( square root of 9+2 square root of 2)

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