To solve the equation [tex]\(-x^{\frac{3}{2}} = -27\)[/tex], follow these steps:
1. Remove the negative sign: We can multiply both sides of the equation by [tex]\(-1\)[/tex] to simplify it:
[tex]\[
x^{\frac{3}{2}} = 27
\][/tex]
2. Isolate [tex]\(x\)[/tex]: To solve for [tex]\(x\)[/tex], we need to undo the exponentiation of [tex]\(\frac{3}{2}\)[/tex]. We can do this by raising both sides of the equation to the reciprocal of [tex]\(\frac{3}{2}\)[/tex], which is [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = 27^{\frac{2}{3}}
\][/tex]
This simplifies to:
[tex]\[
x = 27^{\frac{2}{3}}
\][/tex]
3. Evaluate [tex]\(27^{\frac{2}{3}}\)[/tex]: To find [tex]\(27^{\frac{2}{3}}\)[/tex], we recognize that 27 can be written as [tex]\(3^3\)[/tex]. Therefore,
[tex]\[
27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} = 3^{3 \cdot \frac{2}{3}} = 3^2 = 9
\][/tex]
Thus, the value of [tex]\(x\)[/tex] that satisfies the equation is [tex]\(9\)[/tex].
So, the correct answer to the given equation is:
[tex]\[ \boxed{9} \][/tex]
