(x^ a+b) ^ (a-b) / (x^a-2b) ^ (a+2b)
To simplify the expression (x^(a+b))^(a-b) / (x^a-2b)^(a+2b), we can use the properties of exponents.
First, let's focus on the numerator: (x^(a+b))^(a-b).
By the exponentiation rule, when we raise a power to another power, we multiply the exponents. So, (x^(a+b))^(a-b) simplifies to x^((a+b) * (a-b)).
Next, let's simplify the denominator: (x^a-2b)^(a+2b).
Again, using the exponentiation rule, we get (x^a-2b)^(a+2b) = x^(a * (a+2b)) - 2b * (a+2b).
Now, we have x^((a+b) * (a-b)) / (x^(a * (a+2b)) - 2b * (a+2b)).
Finally, we can simplify further by applying the quotient rule for exponents. According to the quotient rule, when we divide two numbers with the same base, we subtract the exponents. So, the expression becomes x^((a+b) * (a-b) - a * (a+2b) + 2b * (a+2b)).
Therefore, the simplified expression is x^((a² - b²) - a² - 2ab + 2ab + 4b²).
Simplifying further, we get x^(-b² + 4b²) = x^(3b²).
Please note that this is just a simplification of the expression, and if there are any specific values assigned to the variables a, b, and x, you can substitute those values to get the final result.