Solve for a (x^2)/( cube root of x)=x^a

$\frac{x^{2}}{\sqrt[3]{x}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{x^{2}}{\sqrt[3]{x}}$ .

$x^{a} = \frac{x^{2}}{\sqrt[3]{x}}$

Step 2

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

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

Step 3

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

$x^{a} = x^{2} x^{- \frac{1}{3}}$

Step 4

Multiply $x^{2}$ by $x^{- \frac{1}{3}}$ by adding the exponents.

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

$x^{a} = x^{2 - \frac{1}{3}}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{a} = x^{2 \cdot \frac{3}{3} - \frac{1}{3}}$

Combine $2$ and $\frac{3}{3}$ .

$x^{a} = x^{\frac{2 \cdot 3}{3} - \frac{1}{3}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{2 \cdot 3 - 1}{3}}$

Simplify the numerator.

Multiply $2$ by $3$ .

$x^{a} = x^{\frac{6 - 1}{3}}$

Subtract $1$ from $6$ .

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

Step 5

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = 1.\overline{6}$

Mixed Number Form:

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