Simplify the Radical Expression fourth root of 3( fourth root of 2+ fourth root of 4)

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

Step 1

Simplify each term.

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

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

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

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

Pull terms out from under the radical.

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

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 2

Apply the distributive property.

$\sqrt[4]{3} \sqrt[4]{2} + \sqrt[4]{3} \sqrt{2}$

Step 3

Combine using the product rule for radicals.

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

Step 4

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

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

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

$\sqrt[4]{3 \cdot 2} + \sqrt[4]{3} \cdot 2^{\frac{1}{2}}$

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

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

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

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

Combine using the product rule for radicals.

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

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

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

Multiply $3$ by $4$ .

$\sqrt[4]{3 \cdot 2} + \sqrt[4]{12}$

Step 5

Multiply $3$ by $2$ .

$\sqrt[4]{6} + \sqrt[4]{12}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[4]{6} + \sqrt[4]{12}$

Decimal Form:

$3.42629429 \dots$