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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $3$ .

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

Divide $2$ by $2$ .

$x^{a} = x^{1}$

Step 5

Simplify $x^{1}$ .

$x^{a} = x$

Step 6

Create equivalent expressions in the equation that all have equal bases.

$x^{a} = x^{1}$

Step 7

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

$a = 1$