The value of `(6^(n) xx 2^(2n) xx 3^(3n))/(30^(n) xx 3^(2n) xx 2^(3n))` is equal to

Options

A. 1
B. `(0.3)^(n)`
C. `3^(-n)`
D. `3^(5n)`

Answer

To solve the expression \(\frac{6^n \times 2^{2n} \times 3^{3n}}{30^n \times 3^{2n} \times 2^{3n}}\), we will simplify it step by step.

### Step 1: Rewrite the base numbers
First, let's express \(30\) in terms of its prime factors:
\[ 30 = 2 \times 3 \times 5 \]
Thus, we can rewrite \(30^n\) as:
\[ 30^n = (2 \times 3 \times 5)^n = 2^n \times 3^n \times 5^n \]

### Step 2: Substitute back into the expression
Now we can substitute this back into the original expression:
\[ \frac{6^n \times 2^{2n} \times 3^{3n}}{2^n \times 3^n \times 5^n \times 3^{2n} \times 2^{3n}} \]

### Step 3: Combine the terms in the denominator
Next, we can combine the terms in the denominator:
\[ 2^n \times 2^{3n} = 2^{n + 3n} = 2^{4n} \]
\[ 3^n \times 3^{2n} = 3^{n + 2n} = 3^{3n} \]
Thus, the denominator becomes:
\[ 2^{4n} \times 3^{3n} \times 5^n \]

### Step 4: Rewrite the expression
Now, we can rewrite the entire expression:
\[ \frac{6^n \times 2^{2n} \times 3^{3n}}{2^{4n} \times 3^{3n} \times 5^n} \]

### Step 5: Substitute \(6^n\) in terms of its prime factors
Now, express \(6\) in terms of its prime factors:
\[ 6 = 2 \times 3 \]
Thus, we can rewrite \(6^n\) as:
\[ 6^n = (2 \times 3)^n = 2^n \times 3^n \]
Substituting this back gives:
\[ \frac{2^n \times 3^n \times 2^{2n} \times 3^{3n}}{2^{4n} \times 3^{3n} \times 5^n} \]

### Step 6: Combine the terms in the numerator
Now, we can combine the terms in the numerator:
\[ 2^n \times 2^{2n} = 2^{n + 2n} = 2^{3n} \]
\[ 3^n \times 3^{3n} = 3^{n + 3n} = 3^{4n} \]
Thus, the numerator becomes:
\[ 2^{3n} \times 3^{4n} \]

### Step 7: Rewrite the expression again
Now we can rewrite the expression:
\[ \frac{2^{3n} \times 3^{4n}}{2^{4n} \times 3^{3n} \times 5^n} \]

### Step 8: Simplify the expression
Now we can simplify:
\[ \frac{2^{3n}}{2^{4n}} = 2^{3n - 4n} = 2^{-n} \]
\[ \frac{3^{4n}}{3^{3n}} = 3^{4n - 3n} = 3^{n} \]
Thus, the expression simplifies to:
\[ \frac{2^{-n} \times 3^{n}}{5^n} = \frac{3^n}{2^n \times 5^n} \]

### Final Result
The final simplified expression is:
\[ \frac{3^n}{10^n} \]