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 $x^{2a^2-b^2}$.**