Evaluate : `(2^(n)xx6^(m+1)xx10^(m-n)xx15^(m+n-2))/(4^(m)xx3^(2m+n)xx25^(m-1))`

Answer

To evaluate the expression

\[ \frac{2^n \times 6^{m+1} \times 10^{m-n} \times 15^{m+n-2}}{4^m \times 3^{2m+n} \times 25^{m-1}}, \]

we will follow these steps:

### Step 1: Rewrite the bases in terms of their prime factors

1. **Rewrite \(6\), \(10\), \(15\), \(4\), and \(25\)**:
- \(6 = 2 \times 3\)
- \(10 = 2 \times 5\)
- \(15 = 3 \times 5\)
- \(4 = 2^2\)
- \(25 = 5^2\)

Now, substituting these into the expression gives:

\[ \frac{2^n \times (2 \times 3)^{m+1} \times (2 \times 5)^{m-n} \times (3 \times 5)^{m+n-2}}{(2^2)^m \times 3^{2m+n} \times (5^2)^{m-1}}. \]

### Step 2: Simplify the expression

2. **Expand the terms**:
- \(6^{m+1} = (2 \times 3)^{m+1} = 2^{m+1} \times 3^{m+1}\)
- \(10^{m-n} = (2 \times 5)^{m-n} = 2^{m-n} \times 5^{m-n}\)
- \(15^{m+n-2} = (3 \times 5)^{m+n-2} = 3^{m+n-2} \times 5^{m+n-2}\)

Now, substituting these expansions into the expression gives:

\[ \frac{2^n \times 2^{m+1} \times 3^{m+1} \times 2^{m-n} \times 5^{m-n} \times 3^{m+n-2} \times 5^{m+n-2}}{(2^2)^m \times 3^{2m+n} \times (5^2)^{m-1}}. \]

### Step 3: Combine like bases

3. **Combine the powers of like bases in the numerator**:
- For \(2\): \(2^{n + (m+1) + (m-n)} = 2^{2m + 1}\)
- For \(3\): \(3^{(m+1) + (m+n-2)} = 3^{2m + n - 1}\)
- For \(5\): \(5^{(m-n) + (m+n-2)} = 5^{2m - 2}\)

So the numerator simplifies to:

\[ 2^{2m + 1} \times 3^{2m + n - 1} \times 5^{2m - 2}. \]

### Step 4: Simplify the denominator

4. **Rewrite the denominator**:
- \(4^m = (2^2)^m = 2^{2m}\)
- \(25^{m-1} = (5^2)^{m-1} = 5^{2m - 2}\)

Thus, the denominator becomes:

\[ 2^{2m} \times 3^{2m+n} \times 5^{2m - 2}. \]

### Step 5: Write the complete expression

Now, we can write the complete expression as:

\[ \frac{2^{2m+1} \times 3^{2m+n-1} \times 5^{2m-2}}{2^{2m} \times 3^{2m+n} \times 5^{2m-2}}. \]

### Step 6: Cancel out the common terms

5. **Cancel out the common terms**:
- For \(2\): \(2^{2m + 1 - 2m} = 2^1 = 2\)
- For \(3\): \(3^{(2m + n - 1) - (2m + n)} = 3^{-1} = \frac{1}{3}\)
- For \(5\): \(5^{(2m - 2) - (2m - 2)} = 5^0 = 1\)

Thus, the expression simplifies to:

\[ \frac{2}{3}. \]

### Final Answer

Therefore, the final answer is:

\[ \frac{2}{3}. \]