Solve for a ((x^(1/3))^5)/( cube root of x^4)=x^a

$\frac{{(x^{\frac{1}{3}})}^{5}}{\sqrt[3]{x^{4}}} = x^{a}$

Rewrite the equation as $x^{a} = \frac{{(x^{\frac{1}{3}})}^{5}}{\sqrt[3]{x^{4}}}$ .

$x^{a} = \frac{{(x^{\frac{1}{3}})}^{5}}{\sqrt[3]{x^{4}}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{4}}$ as $x^{\frac{4}{3}}$ .

$x^{a} = \frac{{(x^{\frac{1}{3}})}^{5}}{x^{\frac{4}{3}}}$

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

$x^{a} = \frac{x^{\frac{1}{3} \cdot 5}}{x^{\frac{4}{3}}}$

Move $x^{\frac{4}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$x^{a} = x^{\frac{1}{3} \cdot 5} x^{- \frac{4}{3}}$

Multiply $x^{\frac{1}{3} \cdot 5}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

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

$x^{a} = x^{\frac{1}{3} \cdot 5 - \frac{4}{3}}$

Combine $\frac{1}{3}$ and $5$ .

$x^{a} = x^{\frac{5}{3} - \frac{4}{3}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{5 - 4}{3}}$

Subtract $4$ from $5$ .

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

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = 0.\overline{3}$