If `(sqrt(3))^(5) xx 9^(2) =3^(n) xx 3sqrt(3)`, then what is the value of n?

Answer

To solve the equation \((\sqrt{3})^{5} \times 9^{2} = 3^{n} \times 3\sqrt{3}\), we will follow these steps:

### Step 1: Rewrite the terms in the equation using exponents
We know that \(\sqrt{3} = 3^{1/2}\) and \(9 = 3^{2}\). Thus, we can rewrite the left-hand side of the equation:
\[ (\sqrt{3})^{5} = (3^{1/2})^{5} = 3^{(1/2) \times 5} = 3^{5/2} \]
\[ 9^{2} = (3^{2})^{2} = 3^{2 \times 2} = 3^{4} \]
Now, substituting these back into the equation gives:
\[ 3^{5/2} \times 3^{4} \]

### Step 2: Combine the exponents on the left-hand side
Using the property of exponents \(a^{m} \times a^{n} = a^{m+n}\), we can combine the exponents:
\[ 3^{5/2 + 4} = 3^{5/2 + 8/2} = 3^{(5 + 8)/2} = 3^{13/2} \]

### Step 3: Rewrite the right-hand side
Now, let's rewrite the right-hand side of the equation:
\[ 3\sqrt{3} = 3^{1} \times 3^{1/2} = 3^{1 + 1/2} = 3^{3/2} \]
Thus, the right-hand side becomes:
\[ 3^{n} \times 3^{3/2} = 3^{n + 3/2} \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ \frac{13}{2} = n + \frac{3}{2} \]

### Step 5: Solve for \(n\)
To isolate \(n\), we subtract \(\frac{3}{2}\) from both sides:
\[ \frac{13}{2} - \frac{3}{2} = n \]
\[ \frac{10}{2} = n \]
\[ n = 5 \]

### Final Answer
The value of \(n\) is \(5\).

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