`((2^(m))/(2^(n)))^(t)xx((2^(n))/(2^(t)))^(m)xx((2^(t))/(2^(m)))^(n)` is equal to

Options

A. 1
B. 2
C. `(1)/(2)`
D. 0

Answer

To solve the expression \(\left(\frac{2^m}{2^n}\right)^t \times \left(\frac{2^n}{2^t}\right)^m \times \left(\frac{2^t}{2^m}\right)^n\), we can follow these steps:

### Step 1: Simplify Each Term
We can simplify each term using the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\).

1. **First Term:**
\[ \left(\frac{2^m}{2^n}\right)^t = \left(2^{m-n}\right)^t = 2^{(m-n)t} \]

2. **Second Term:**
\[ \left(\frac{2^n}{2^t}\right)^m = \left(2^{n-t}\right)^m = 2^{(n-t)m} \]

3. **Third Term:**
\[ \left(\frac{2^t}{2^m}\right)^n = \left(2^{t-m}\right)^n = 2^{(t-m)n} \]

### Step 2: Combine the Terms
Now, we can combine all the simplified terms:
\[ 2^{(m-n)t} \times 2^{(n-t)m} \times 2^{(t-m)n} \]

Using the property \(a^m \times a^n = a^{m+n}\), we can combine the exponents:
\[ 2^{(m-n)t + (n-t)m + (t-m)n} \]

### Step 3: Simplify the Exponent
Now, we need to simplify the exponent:
\[ (m-n)t + (n-t)m + (t-m)n \]

Expanding each term:
1. \((m-n)t = mt - nt\)
2. \((n-t)m = nm - mt\)
3. \((t-m)n = tn - mn\)

Combining these, we have:
\[ mt - nt + nm - mt + tn - mn \]

Notice that \(mt\) and \(-mt\) cancel each other, as do \(nt\) and \(-nt\), and \(nm\) and \(-mn\):
\[ 0 \]

### Step 4: Final Result
Thus, we have:
\[ 2^0 = 1 \]

So, the final answer is:
\[ \boxed{1} \]

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