Solve for a x^3 cube root of x=x^a

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $3$ .

$x^{a} = x^{\frac{9 + 1}{3}}$

Add $9$ and $1$ .

$x^{a} = x^{\frac{10}{3}}$

Step 4

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

$a = \frac{10}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$a = \frac{10}{3}$

Decimal Form:

$a = 3.\overline{3}$

Mixed Number Form:

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