Simplify : `((x^a)/(x^b) )^(a+b) div ((x^a)/( x^(a-b) ))^(a^(2) //b)`

Answer

To simplify the expression \(\frac{(x^a/x^b)^{(a+b)}}{(x^a/x^{(a-b)})^{(a^2/b)}}\), we will follow these steps:

### Step 1: Simplify the numerator
The numerator is \((x^a/x^b)^{(a+b)}\).
Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify \(x^a/x^b\) as follows:
\[ x^a/x^b = x^{a-b} \]
Thus, the numerator becomes:
\[ (x^{a-b})^{(a+b)} \]

### Step 2: Apply the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ (x^{a-b})^{(a+b)} = x^{(a-b)(a+b)} \]

### Step 3: Simplify the denominator
The denominator is \((x^a/x^{(a-b)})^{(a^2/b)}\).
Again, using the property of exponents:
\[ x^a/x^{(a-b)} = x^{a - (a-b)} = x^{a - a + b} = x^b \]
Thus, the denominator becomes:
\[ (x^b)^{(a^2/b)} \]

### Step 4: Apply the power of a power property again
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ (x^b)^{(a^2/b)} = x^{b \cdot (a^2/b)} = x^{a^2} \]

### Step 5: Combine the results
Now we can rewrite the entire expression:
\[ \frac{x^{(a-b)(a+b)}}{x^{a^2}} \]
Using the property \(\frac{a^m}{a^n} = a^{m-n}\), we have:
\[ x^{(a-b)(a+b) - a^2} \]

### Step 6: Simplify the exponent
Now we need to simplify the exponent:
\[ (a-b)(a+b) - a^2 \]
Using the difference of squares:
\[ (a-b)(a+b) = a^2 - b^2 \]
Thus, we have:
\[ a^2 - b^2 - a^2 = -b^2 \]

### Final Result
Therefore, the expression simplifies to:
\[ x^{-b^2} \]