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

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

Rewrite the equation as $x^{a} = x^{\frac{1}{3}} \sqrt{x^{5}}$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{5}}$ as $x^{\frac{5}{2}}$ .

$x^{a} = x^{\frac{1}{3}} x^{\frac{5}{2}}$

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{5}{2}}$ by adding the exponents.

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

$x^{a} = x^{\frac{1}{3} + \frac{5}{2}}$

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{a} = x^{\frac{1}{3} \cdot \frac{2}{2} + \frac{5}{2}}$

To write $\frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{a} = x^{\frac{1}{3} \cdot \frac{2}{2} + \frac{5}{2} \cdot \frac{3}{3}}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$x^{a} = x^{\frac{2}{3 \cdot 2} + \frac{5}{2} \cdot \frac{3}{3}}$

Multiply $3$ by $2$ .

$x^{a} = x^{\frac{2}{6} + \frac{5}{2} \cdot \frac{3}{3}}$

Multiply $\frac{5}{2}$ by $\frac{3}{3}$ .

$x^{a} = x^{\frac{2}{6} + \frac{5 \cdot 3}{2 \cdot 3}}$

Multiply $2$ by $3$ .

$x^{a} = x^{\frac{2}{6} + \frac{5 \cdot 3}{6}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{2 + 5 \cdot 3}{6}}$

Simplify the numerator.

Multiply $5$ by $3$ .

$x^{a} = x^{\frac{2 + 15}{6}}$

Add $2$ and $15$ .

$x^{a} = x^{\frac{17}{6}}$

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

$a = \frac{17}{6}$

The result can be shown in multiple forms.

Exact Form:

$a = \frac{17}{6}$

Decimal Form:

$a = 2.8\overline{3}$

Mixed Number Form:

$a = 2 \frac{5}{6}$