If `3sqrt(3)xx3^(3)-:3^(-3//2)=3^(a+2)` , then a =

Options

A. `2`
B. `1//2`
C. `4`
D. `0`

Answer

To solve the equation \( 3\sqrt{3} \times 3^3 \div 3^{-3/2} = 3^{a+2} \), we will simplify the left-hand side step by step and then equate the exponents to find the value of \( a \).

### Step 1: Rewrite the equation
We start with the equation:
\[ 3\sqrt{3} \times 3^3 \div 3^{-3/2} = 3^{a+2} \]

### Step 2: Convert \( \sqrt{3} \) to exponent form
We know that \( \sqrt{3} = 3^{1/2} \). Therefore, we can rewrite \( 3\sqrt{3} \) as:
\[ 3\sqrt{3} = 3^1 \times 3^{1/2} = 3^{1 + 1/2} = 3^{3/2} \]

### Step 3: Substitute back into the equation
Now we can substitute this back into our equation:
\[ 3^{3/2} \times 3^3 \div 3^{-3/2} = 3^{a+2} \]

### Step 4: Combine the terms in the numerator
Using the property of exponents \( a^m \times a^n = a^{m+n} \), we combine the terms in the numerator:
\[ 3^{3/2 + 3} = 3^{3/2 + 6/2} = 3^{9/2} \]

### Step 5: Rewrite the division
Now, we rewrite the division:
\[ 3^{9/2} \div 3^{-3/2} = 3^{9/2 - (-3/2)} = 3^{9/2 + 3/2} = 3^{12/2} = 3^6 \]

### Step 6: Set the exponents equal
Now we have:
\[ 3^6 = 3^{a+2} \]
Since the bases are the same, we can equate the exponents:
\[ 6 = a + 2 \]

### Step 7: Solve for \( a \)
To find \( a \), we subtract 2 from both sides:
\[ a = 6 - 2 = 4 \]

### Conclusion
Thus, the value of \( a \) is:
\[ \boxed{4} \]