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

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

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

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

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

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

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

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

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $- 2$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 2$ by $2$ .

$x^{a} = x^{\frac{1 - 4}{2}}$

Subtract $4$ from $1$ .

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

Move the negative in front of the fraction.

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

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

$a = - \frac{3}{2}$

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{3}{2}$

Decimal Form:

$a = - 1.5$

Mixed Number Form:

$a = - 1 \frac{1}{2}$