Solve for a ( fifth root of x^4)/(x^(1/2))=x^a

$\frac{\sqrt[5]{x^{4}}}{x^{\frac{1}{2}}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{\sqrt[5]{x^{4}}}{x^{\frac{1}{2}}}$ .

$x^{a} = \frac{\sqrt[5]{x^{4}}}{x^{\frac{1}{2}}}$

Step 2

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

$x^{a} = \frac{x^{\frac{4}{5}}}{x^{\frac{1}{2}}}$

Step 3

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

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

Step 4

Multiply $x^{\frac{4}{5}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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

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

To write $\frac{4}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

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

Multiply $\frac{4}{5}$ by $\frac{2}{2}$ .

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

Multiply $5$ by $2$ .

$x^{a} = x^{\frac{4 \cdot 2}{10} - \frac{1}{2} \cdot \frac{5}{5}}$

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

$x^{a} = x^{\frac{4 \cdot 2}{10} - \frac{5}{2 \cdot 5}}$

Multiply $2$ by $5$ .

$x^{a} = x^{\frac{4 \cdot 2}{10} - \frac{5}{10}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{4 \cdot 2 - 5}{10}}$

Simplify the numerator.

Multiply $4$ by $2$ .

$x^{a} = x^{\frac{8 - 5}{10}}$

Subtract $5$ from $8$ .

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

Step 5

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = 0.3$