Simplify and express as positive indices:
(a−2b)12×(ab−3)13(a^{-2}b)^{\dfrac{1}{2}} \times (ab^{-3})^{\dfrac{1}{3}}(a−2b)21×(ab−3)31
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(a−2b)12×(ab−3)13=(a−2×12b12)×(a13b−3×13)=(a−1b12)×(a13b−1)=(a−1+13b12+(−1))=(a−33+13b12+−22)=(a−3+13b1−22)=(a−23b−12)=1a23b12(a^{-2}b)^{\dfrac{1}{2}} \times (ab^{-3})^{\dfrac{1}{3}}\\[1em] = \Big(a^{-2\times\dfrac{1}{2}}b^{\dfrac{1}{2}}\Big) \times \Big(a^{\dfrac{1}{3}}b^{-3\times\dfrac{1}{3}}\Big)\\[1em] = \Big(a^{-1}b^{\dfrac{1}{2}}\Big) \times \Big(a^{\dfrac{1}{3}}b^{-1}\Big)\\[1em] = \Big(a^{-1 + \dfrac{1}{3}}b^{\dfrac{1}{2} + (-1)}\Big)\\[1em] = \Big(a^{\dfrac{-3}{3} + \dfrac{1}{3}}b^{\dfrac{1}{2} + \dfrac{-2}{2}}\Big)\\[1em] = \Big(a^{\dfrac{-3+1}{3}}b^{\dfrac{1-2}{2}}\Big)\\[1em] = \Big(a^{\dfrac{-2}{3}}b^{\dfrac{-1}{2}}\Big)\\[1em] = \dfrac{1}{a^{\dfrac{2}{3}}b^{\dfrac{1}{2}}}(a−2b)21×(ab−3)31=(a−2×21b21)×(a31b−3×31)=(a−1b21)×(a31b−1)=(a−1+31b21+(−1))=(a3−3+31b21+2−2)=(a3−3+1b21−2)=(a3−2b2−1)=a32b211
(a−2b)12×(ab−3)13=1a23b12(a^{-2}b)^{\dfrac{1}{2}} \times (ab^{-3})^{\dfrac{1}{3}} = \dfrac{1}{a^{\dfrac{2}{3}}b^{\dfrac{1}{2}}}(a−2b)21×(ab−3)31=a32b211
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