Solve for a (x^4)^(1/3) cube root of x=x^a

${(x^{4})}^{\frac{1}{3}} \sqrt[3]{x} = x^{a}$

Rewrite the equation as $x^{a} = {(x^{4})}^{\frac{1}{3}} \sqrt[3]{x}$ .

$x^{a} = {(x^{4})}^{\frac{1}{3}} \sqrt[3]{x}$

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

$x^{a} = {(x^{4})}^{\frac{1}{3}} x^{\frac{1}{3}}$

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

$x^{a} = x^{4 (\frac{1}{3})} x^{\frac{1}{3}}$

Multiply $x^{4 (\frac{1}{3})}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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

$x^{a} = x^{4 (\frac{1}{3}) + \frac{1}{3}}$

Combine $4$ and $\frac{1}{3}$ .

$x^{a} = x^{\frac{4}{3} + \frac{1}{3}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{4 + 1}{3}}$

Add $4$ and $1$ .

$x^{a} = x^{\frac{5}{3}}$

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$a = \frac{5}{3}$

The result can be shown in multiple forms.

Exact Form:

$a = \frac{5}{3}$

Decimal Form:

$a = 1.\overline{6}$

Mixed Number Form:

$a = 1 \frac{2}{3}$