Which expression is equivalent to $$ \sqrt{9 z^{3}} $$ ? $$ 3 z^{\frac{2}{3}} \frac{3 z^{3}}{2} $$ $$ 3 z^{3}-\frac{3}{2} $$

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$$ 3 z^{\frac{2}{3}} $$

To determine which expression is equivalent to $$ \sqrt{9 z^{3}} $$, let's simplify the given expression step by step.

First, we can simplify the square root of $$ 9 z^{3} $$ by breaking it down into two parts: the square root of the coefficient and the square root of the variable term.

1. Square root of the coefficient:
   $$ \sqrt{9} = 3 $$

2. Square root of the variable term:
   $$ \sqrt{z^{3}} = z^{\frac{3}{2}} $$

Now, we can combine these two parts to get the simplified expression:
$$ \sqrt{9 z^{3}} = 3 z^{\frac{3}{2}} $$

Let's compare this simplified expression with the given options:

1. $$ 3 z^{\frac{2}{3}} \frac{3 z^{3}}{2} $$
   This expression is not equivalent to $$ 3 z^{\frac{3}{2}} $$ because the exponents do not match.

2. $$ 3 z^{3}-\frac{3}{2} $$
   This expression is not equivalent to $$ 3 z^{\frac{3}{2}} $$ because it has a subtraction and a different exponent.

3. $$ 3 z^{\frac{2}{3}} $$
   This expression is not equivalent to $$ 3 z^{\frac{3}{2}} $$ because the exponents do not match.

Therefore, none of the given options are equivalent to $$ \sqrt{9 z^{3}} $$. The correct simplified expression is $$ 3 z^{\frac{3}{2}} $$.
The correct simplified expression is $$ 3 z^{\frac{3}{2}} $$.

Which expression is equivalent to ?

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