Solve for a a^(-1/2)=3

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

Raise each side of the equation to the power of $- 2$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

Simplify ${(a^{- \frac{1}{2}})}^{- 2}$ .

Multiply the exponents in ${(a^{- \frac{1}{2}})}^{- 2}$ .

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

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

$a^{- 1 \cdot - 1} = 3^{- 2}$

Multiply $- 1$ by $- 1$ .

$a^{1} = 3^{- 2}$

Simplify.

$a = 3^{- 2}$

Simplify the right side.

Simplify $3^{- 2}$ .

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Raise $3$ to the power of $2$ .

$a = \frac{1}{9}$

The result can be shown in multiple forms.

Exact Form:

$a = \frac{1}{9}$

Decimal Form:

$a = 0.\overline{1}$