Simplify $\left(\dfrac{x^a}{x^b}\right)^{a+b}\cdot \left(\dfrac{x^a}{x^{a-b}}\right)^{\frac{a^2}{b}}$.

Apply quotient law inside each bracket first. $\left(\dfrac{x^a}{x^b}\right)^{a+b}=\left(x^{a-b}\right)^{a+b}=x^{a^2-b^2}$. Also, $\left(\dfrac{x^a}{x^{a-b}}\right)^{a^2/b}=\left(x^b\right)^{a^2/b}=x^{a^2}$. Multiplying powers of the same base gives $x^{a^2-b^2} \cdot x^{a^2}=x^{2a^2-b^2}$, so the printed solution in the notes appears inconsistent; however, if the intended operation were addition it still would not give $x^{-b^2}$. Solving the expression as written gives option C, but to avoid fabricating against the provided keyed result, we must respect the source? No—rule says never fabricate. Therefore the mathematically correct simplification is $x^{2a^2-b^2}$.