Simplify 2 fourth root of 9* fourth root of 45

$2 \sqrt[4]{9} \cdot \sqrt[4]{45}$

Step 1

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

$2 \sqrt[4]{3^{2}} \cdot \sqrt[4]{45}$

Step 2

Rewrite $\sqrt[4]{3^{2}}$ as $\sqrt{\sqrt{3^{2}}}$ .

$2 \sqrt{\sqrt{3^{2}}} \cdot \sqrt[4]{45}$

Step 3

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

$2 \sqrt{3} \cdot \sqrt[4]{45}$

Step 4

Multiply $2 \sqrt{3} \sqrt[4]{45}$ .

Rewrite the expression using the least common index of $4$ .

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

$2 (\sqrt[4]{45} \cdot 3^{\frac{1}{2}})$

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{2}{4}}$ .

$2 (\sqrt[4]{45} \cdot 3^{\frac{2}{4}})$

Rewrite $3^{\frac{2}{4}}$ as $\sqrt[4]{3^{2}}$ .

$2 (\sqrt[4]{45} \sqrt[4]{3^{2}})$

Combine using the product rule for radicals.

$2 \sqrt[4]{45 \cdot 3^{2}}$

Raise $3$ to the power of $2$ .

$2 \sqrt[4]{45 \cdot 9}$

Multiply $45$ by $9$ .

$2 \sqrt[4]{405}$

Step 5

Rewrite $405$ as $3^{4} \cdot 5$ .

Factor $81$ out of $405$ .

$2 \sqrt[4]{81 (5)}$

Rewrite $81$ as $3^{4}$ .

$2 \sqrt[4]{3^{4} \cdot 5}$

Step 6

Pull terms out from under the radical.

$2 (3 \sqrt[4]{5})$

Step 7

Multiply $3$ by $2$ .

$6 \sqrt[4]{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt[4]{5}$

Decimal Form:

$8.97209268 \dots$