Solve for a (x^(1/3))^2 cube root of x=x^a

${(x^{\frac{1}{3}})}^{2} \sqrt[3]{x} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = {(x^{\frac{1}{3}})}^{2} \sqrt[3]{x}$ .

$x^{a} = {(x^{\frac{1}{3}})}^{2} \sqrt[3]{x}$

Step 2

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

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

Step 3

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

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

Step 4

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

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

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

Combine $\frac{1}{3}$ and $2$ .

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

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

Divide $3$ by $3$ .

$x^{a} = x^{1}$

Step 5

Simplify $x^{1}$ .

$x^{a} = x$

Step 6

Create equivalent expressions in the equation that all have equal bases.

$x^{a} = x^{1}$

Step 7

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

$a = 1$