`9^(3//2) div (243)^(-2//3)` simplifies to :

Options

A. `3^(10//3)`
B. `3^(19//3)`
C. `3^(1//3)`
D. `3^(19)`

Answer

To simplify the expression \( 9^{\frac{3}{2}} \div (243)^{-\frac{2}{3}} \), we will follow these steps:

### Step 1: Rewrite the bases in terms of powers of 3
We know that:
- \( 9 = 3^2 \)
- \( 243 = 3^5 \)

So we can rewrite the expression as:
\[ (3^2)^{\frac{3}{2}} \div (3^5)^{-\frac{2}{3}} \]

### Step 2: Apply the power of a power property
Using the property \( (a^m)^n = a^{m \cdot n} \), we simplify both parts:
\[ (3^2)^{\frac{3}{2}} = 3^{2 \cdot \frac{3}{2}} = 3^3 \]
\[ (3^5)^{-\frac{2}{3}} = 3^{5 \cdot -\frac{2}{3}} = 3^{-\frac{10}{3}} \]

So now we have:
\[ 3^3 \div 3^{-\frac{10}{3}} \]

### Step 3: Apply the division of powers property
Using the property \( a^m \div a^n = a^{m-n} \), we combine the powers:
\[ 3^3 \div 3^{-\frac{10}{3}} = 3^{3 - (-\frac{10}{3})} = 3^{3 + \frac{10}{3}} \]

### Step 4: Simplify the exponent
To add \( 3 \) and \( \frac{10}{3} \), we first convert \( 3 \) into a fraction with a common denominator:
\[ 3 = \frac{9}{3} \]
Now we can add:
\[ \frac{9}{3} + \frac{10}{3} = \frac{19}{3} \]

### Step 5: Write the final result
Thus, we have:
\[ 3^{\frac{19}{3}} \]

### Final Answer
The simplified expression is:
\[ 3^{\frac{19}{3}} \]

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