Solve for a ( fourth root of x^5)/(x^(1/6))=x^a

$\frac{\sqrt[4]{x^{5}}}{x^{\frac{1}{6}}} = x^{a}$

Rewrite the equation as $x^{a} = \frac{\sqrt[4]{x^{5}}}{x^{\frac{1}{6}}}$ .

$x^{a} = \frac{\sqrt[4]{x^{5}}}{x^{\frac{1}{6}}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{5}}$ as $x^{\frac{5}{4}}$ .

$x^{a} = \frac{x^{\frac{5}{4}}}{x^{\frac{1}{6}}}$

Move $x^{\frac{1}{6}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$x^{a} = x^{\frac{5}{4}} x^{- \frac{1}{6}}$

Multiply $x^{\frac{5}{4}}$ by $x^{- \frac{1}{6}}$ by adding the exponents.

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

$x^{a} = x^{\frac{5}{4} - \frac{1}{6}}$

To write $\frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{a} = x^{\frac{5}{4} \cdot \frac{3}{3} - \frac{1}{6}}$

To write $- \frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{a} = x^{\frac{5}{4} \cdot \frac{3}{3} - \frac{1}{6} \cdot \frac{2}{2}}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{4}$ by $\frac{3}{3}$ .

$x^{a} = x^{\frac{5 \cdot 3}{4 \cdot 3} - \frac{1}{6} \cdot \frac{2}{2}}$

Multiply $4$ by $3$ .

$x^{a} = x^{\frac{5 \cdot 3}{12} - \frac{1}{6} \cdot \frac{2}{2}}$

Multiply $\frac{1}{6}$ by $\frac{2}{2}$ .

$x^{a} = x^{\frac{5 \cdot 3}{12} - \frac{2}{6 \cdot 2}}$

Multiply $6$ by $2$ .

$x^{a} = x^{\frac{5 \cdot 3}{12} - \frac{2}{12}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{5 \cdot 3 - 2}{12}}$

Simplify the numerator.

Multiply $5$ by $3$ .

$x^{a} = x^{\frac{15 - 2}{12}}$

Subtract $2$ from $15$ .

$x^{a} = x^{\frac{13}{12}}$

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

$a = \frac{13}{12}$

The result can be shown in multiple forms.

Exact Form:

$a = \frac{13}{12}$

Decimal Form:

$a = 1.08\overline{3}$

Mixed Number Form:

$a = 1 \frac{1}{12}$