Question

solve the equation with rational exponents. check all proposed solutions.
$3x^{\frac{9}{2}} - 33 = 0$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is \boxed{}.
(type an exact answer in simplified form. use a comma to separate answers as needed.)
b. the solution set is \\(\varnothing\\).

Explanation:

Step1: Isolate the term with x

Add 33 to both sides of the equation \(3x^{\frac{9}{2}} - 33 = 0\) to get \(3x^{\frac{9}{2}} = 33\).

Step2: Solve for \(x^{\frac{9}{2}}\)

Divide both sides by 3: \(x^{\frac{9}{2}}=\frac{33}{3} = 11\).

Step3: Solve for x

Raise both sides to the power of \(\frac{2}{9}\) to solve for x. Using the property \((a^m)^n=a^{mn}\), we have \(x=(11)^{\frac{2}{9}}\) or \(x=\sqrt[9]{11^{2}}=\sqrt[9]{121}\). We should also check if this solution is valid. Substitute \(x = \sqrt[9]{121}\) back into the original equation:
Left - hand side: \(3(\sqrt[9]{121})^{\frac{9}{2}}-33=3\times121^{\frac{1}{2}} - 33=3\times\sqrt{121}-33=3\times11 - 33=33 - 33 = 0\), which matches the right - hand side.

Answer:

\(\sqrt[9]{121}\) (or \(11^{\frac{2}{9}}\))