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

$\frac{\sqrt[4]{x^{5}}}{x^{3}} = x^{a}$

Rewrite the equation as $x^{a} = \frac{\sqrt[4]{x^{5}}}{x^{3}}$ .

$x^{a} = \frac{\sqrt[4]{x^{5}}}{x^{3}}$

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

$x^{a} = \frac{x^{\frac{5}{4}}}{x^{3}}$

Move $x^{3}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Multiply $x^{\frac{5}{4}}$ by $x^{- 3}$ by adding the exponents.

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

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

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

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

Combine $- 3$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 3$ by $4$ .

$x^{a} = x^{\frac{5 - 12}{4}}$

Subtract $12$ from $5$ .

$x^{a} = x^{\frac{- 7}{4}}$

Move the negative in front of the fraction.

$x^{a} = x^{- \frac{7}{4}}$

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

$a = - \frac{7}{4}$

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{7}{4}$

Decimal Form:

$a = - 1.75$

Mixed Number Form:

$a = - 1 \frac{3}{4}$