Simplify : `((a^(m))/(a^(n)))^(m+n)((a^(n))/(a^(l)))^(n+l)((a^(l))/(a^(m)))^(l+m)`

Answer

To simplify the expression \(\left(\frac{a^m}{a^n}\right)^{m+n} \left(\frac{a^n}{a^l}\right)^{n+l} \left(\frac{a^l}{a^m}\right)^{l+m}\), we can follow these steps:

### Step 1: Rewrite the fractions using the property of exponents
We can use the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\). Thus, we rewrite each fraction:

\[ \frac{a^m}{a^n} = a^{m-n}, \quad \frac{a^n}{a^l} = a^{n-l}, \quad \frac{a^l}{a^m} = a^{l-m} \]

### Step 2: Substitute back into the expression
Now we substitute these back into the original expression:

\[ \left(a^{m-n}\right)^{m+n} \left(a^{n-l}\right)^{n+l} \left(a^{l-m}\right)^{l+m} \]

### Step 3: Apply the power of a power property
Using the property \((a^x)^y = a^{xy}\), we can simplify each term:

\[ a^{(m-n)(m+n)} \cdot a^{(n-l)(n+l)} \cdot a^{(l-m)(l+m)} \]

### Step 4: Combine the exponents
When multiplying powers with the same base, we add the exponents:

\[ a^{(m-n)(m+n) + (n-l)(n+l) + (l-m)(l+m)} \]

### Step 5: Expand each term
Now we will expand each of the terms in the exponent:

1. \((m-n)(m+n) = m^2 - n^2\)
2. \((n-l)(n+l) = n^2 - l^2\)
3. \((l-m)(l+m) = l^2 - m^2\)

So, we have:

\[ m^2 - n^2 + n^2 - l^2 + l^2 - m^2 \]

### Step 6: Simplify the expression
Now, we can see that the terms cancel out:

\[ m^2 - m^2 + n^2 - n^2 + l^2 - l^2 = 0 \]

Thus, we are left with:

\[ a^0 \]

### Step 7: Final simplification
Since any non-zero number raised to the power of 0 is 1, we conclude:

\[ a^0 = 1 \]

### Final Answer:
The simplified expression is \(1\).

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