The value of `(2^(m+1).3^(2m-n).5^(m+n).6^(n))/(6^m.10^(n+2).15^(m))` is equal to

Options

A. 50
B. `1/25`
C. `1/50`
D. `1/5`

Answer

To solve the expression \((2^{(m+1)} \cdot 3^{(2m-n)} \cdot 5^{(m+n)} \cdot 6^{n}) / (6^{m} \cdot 10^{(n+2)} \cdot 15^{m})\), we will simplify it step by step.

### Step 1: Rewrite the bases
First, we rewrite \(6\), \(10\), and \(15\) in terms of their prime factors:
- \(6 = 2 \cdot 3\)
- \(10 = 2 \cdot 5\)
- \(15 = 3 \cdot 5\)

### Step 2: Substitute the prime factorization
Now substitute these into the expression:
\[ 6^m = (2 \cdot 3)^m = 2^m \cdot 3^m \]
\[ 10^{(n+2)} = (2 \cdot 5)^{(n+2)} = 2^{(n+2)} \cdot 5^{(n+2)} \]
\[ 15^m = (3 \cdot 5)^m = 3^m \cdot 5^m \]

### Step 3: Substitute into the original expression
Now we can rewrite the original expression:
\[ \frac{2^{(m+1)} \cdot 3^{(2m-n)} \cdot 5^{(m+n)} \cdot 6^{n}}{6^{m} \cdot 10^{(n+2)} \cdot 15^{m}} = \frac{2^{(m+1)} \cdot 3^{(2m-n)} \cdot 5^{(m+n)} \cdot (2 \cdot 3)^{n}}{(2^m \cdot 3^m) \cdot (2^{(n+2)} \cdot 5^{(n+2)}) \cdot (3^m \cdot 5^m)} \]

### Step 4: Combine the terms in the numerator
The numerator becomes:
\[ 2^{(m+1)} \cdot 3^{(2m-n)} \cdot 5^{(m+n)} \cdot 2^{n} \cdot 3^{n} = 2^{(m+1+n)} \cdot 3^{(2m-n+n)} \cdot 5^{(m+n)} \]
This simplifies to:
\[ 2^{(m+n+1)} \cdot 3^{(2m)} \cdot 5^{(m+n)} \]

### Step 5: Combine the terms in the denominator
The denominator becomes:
\[ (2^m \cdot 3^m) \cdot (2^{(n+2)} \cdot 5^{(n+2)}) \cdot (3^m \cdot 5^m) = 2^{(m+n+2)} \cdot 3^{(2m)} \cdot 5^{(n+2+m)} \]

### Step 6: Write the full expression
Now we have:
\[ \frac{2^{(m+n+1)} \cdot 3^{(2m)} \cdot 5^{(m+n)}}{2^{(m+n+2)} \cdot 3^{(2m)} \cdot 5^{(m+n+2)}} \]

### Step 7: Simplify the expression
Now we can simplify:
1. For \(2\):
\[ 2^{(m+n+1 - (m+n+2))} = 2^{-1} = \frac{1}{2} \]
2. For \(3\):
\[ 3^{(2m - 2m)} = 3^0 = 1 \]
3. For \(5\):
\[ 5^{(m+n - (m+n+2))} = 5^{-2} = \frac{1}{25} \]

### Step 8: Combine the results
Putting it all together:
\[ \frac{1}{2} \cdot 1 \cdot \frac{1}{25} = \frac{1}{50} \]

### Final Answer
The value of the expression is \(\frac{1}{50}\).

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