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

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

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

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

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

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

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

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

Multiply $5$ by $2$ .

$x^{a} = x^{\frac{2}{10} - \frac{1}{2} \cdot \frac{5}{5}}$

Multiply $\frac{1}{2}$ by $\frac{5}{5}$ .

$x^{a} = x^{\frac{2}{10} - \frac{5}{2 \cdot 5}}$

Multiply $2$ by $5$ .

$x^{a} = x^{\frac{2}{10} - \frac{5}{10}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{2 - 5}{10}}$

Subtract $5$ from $2$ .

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

Move the negative in front of the fraction.

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

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = - 0.3$