Simplify ( fourth root of 4)^4

${(\sqrt[4]{4})}^{4}$

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

${\sqrt[4]{2^{2}}}^{4}$

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

${\sqrt{\sqrt{2^{2}}}}^{4}$

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

${\sqrt{2}}^{4}$

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

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

${(2^{\frac{1}{2}})}^{4}$

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

$2^{\frac{1}{2} \cdot 4}$

Combine $\frac{1}{2}$ and $4$ .

$2^{\frac{4}{2}}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$2^{\frac{2 \cdot 2}{2}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$2^{\frac{2 \cdot 2}{2 (1)}}$

Cancel the common factor.

$2^{\frac{2 \cdot 2}{2 \cdot 1}}$

Rewrite the expression.

$2^{\frac{2}{1}}$

Divide $2$ by $1$ .

$2^{2}$

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

$4$