Solve for a (x^5)^(1/4) fourth root of x^3=x^a

${(x^{5})}^{\frac{1}{4}} \sqrt[4]{x^{3}} = x^{a}$

Rewrite the equation as $x^{a} = {(x^{5})}^{\frac{1}{4}} \sqrt[4]{x^{3}}$ .

$x^{a} = {(x^{5})}^{\frac{1}{4}} \sqrt[4]{x^{3}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

$x^{a} = {(x^{5})}^{\frac{1}{4}} x^{\frac{3}{4}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{a} = x^{5 (\frac{1}{4})} x^{\frac{3}{4}}$

Multiply $x^{5 (\frac{1}{4})}$ by $x^{\frac{3}{4}}$ by adding the exponents.

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

$x^{a} = x^{5 (\frac{1}{4}) + \frac{3}{4}}$

Combine $5$ and $\frac{1}{4}$ .

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

Combine the numerators over the common denominator.

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

Add $5$ and $3$ .

$x^{a} = x^{\frac{8}{4}}$

Divide $8$ by $4$ .

$x^{a} = x^{2}$

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

$a = 2$