Problem: Of the following expressions the one equal to a−1b−1a−3−b−3\dfrac{a^{-1} b^{-1}}{a^{-3}-b^{-3}}a−3−b−3a−1b−1 is:
Answer Choices:
A. a2b2b2−a2\dfrac{a^{2} b^{2}}{b^{2}-a^{2}}b2−a2a2b2
B. a2b2b3−a3\dfrac{a^{2} b^{2}}{b^{3}-a^{3}}b3−a3a2b2
C. abb3−a3\dfrac{a b}{b^{3}-a^{3}}b3−a3ab
D. a3−b3ab\dfrac{a^{3}-b^{3}}{a b}aba3−b3
E. a3b3a−b\dfrac{a^{3} b^{3}}{a-b}a−ba3b3 Solution:
a−1b−1a−3−b−3⋅a3b3a3b3=a2b2b3−a3\dfrac{a^{-1} b^{-1}}{a^{-3}-b^{-3}} \cdot \dfrac{a^{3} b^{3}}{a^{3} b^{3}}=\dfrac{a^{2} b^{2}}{b^{3}-a^{3}}a−3−b−3a−1b−1⋅a3b3a3b3=b3−a3a2b2