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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $3$ .

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

Divide $4$ by $2$ .

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

Step 4

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

$a = 2$