Solve for a x^(1/3)x=x^a

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

Rewrite the equation as $x^{a} = x^{\frac{1}{3}} x$ .

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

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

Multiply $x^{\frac{1}{3}}$ by $x$ .

Raise $x$ to the power of $1$ .

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $1$ and $3$ .

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

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

$a = \frac{4}{3}$

The result can be shown in multiple forms.

Exact Form:

$a = \frac{4}{3}$

Decimal Form:

$a = 1.\overline{3}$

Mixed Number Form:

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