Solve for a a^(3/2)=-27

$a^{\frac{3}{2}} = - 27$

Step 1

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(a^{\frac{3}{2}})}^{\frac{2}{3}} = {(- 27)}^{\frac{2}{3}}$

Step 2

Simplify the exponent.

Simplify the left side.

Simplify ${(a^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Multiply the exponents in ${(a^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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

$a^{\frac{3}{2} \cdot \frac{2}{3}} = {(- 27)}^{\frac{2}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$a^{\frac{3}{2} \cdot \frac{2}{3}} = {(- 27)}^{\frac{2}{3}}$

Rewrite the expression.

$a^{\frac{1}{2} \cdot 2} = {(- 27)}^{\frac{2}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$a^{\frac{1}{2} \cdot 2} = {(- 27)}^{\frac{2}{3}}$

Rewrite the expression.

$a^{1} = {(- 27)}^{\frac{2}{3}}$

Simplify.

$a = {(- 27)}^{\frac{2}{3}}$

Simplify the right side.

Simplify ${(- 27)}^{\frac{2}{3}}$ .

Simplify the expression.

Rewrite $- 27$ as ${(- 3)}^{3}$ .

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$a = {(- 3)}^{2}$

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

$a = 9$

Step 3

Exclude the solutions that do not make $a^{\frac{3}{2}} = - 27$ true.

$No solution$