Solve for a (x^(1/2))^3 square root of x^3=x^a

${(x^{\frac{1}{2}})}^{3} \sqrt{x^{3}} = x^{a}$

Rewrite the equation as $x^{a} = {(x^{\frac{1}{2}})}^{3} \sqrt{x^{3}}$ .

$x^{a} = {(x^{\frac{1}{2}})}^{3} \sqrt{x^{3}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$x^{a} = {(x^{\frac{1}{2}})}^{3} x^{\frac{3}{2}}$

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

$x^{a} = x^{\frac{1}{2} \cdot 3} x^{\frac{3}{2}}$

Multiply $x^{\frac{1}{2} \cdot 3}$ by $x^{\frac{3}{2}}$ by adding the exponents.

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

$x^{a} = x^{\frac{1}{2} \cdot 3 + \frac{3}{2}}$

Combine $\frac{1}{2}$ and $3$ .

$x^{a} = x^{\frac{3}{2} + \frac{3}{2}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{3 + 3}{2}}$

Add $3$ and $3$ .

$x^{a} = x^{\frac{6}{2}}$

Divide $6$ by $2$ .

$x^{a} = x^{3}$

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

$a = 3$