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We describe, analyze and compare some combinations of multi-exponentiation algorithms with representations of the exponents. We are especially interested in the case where the inversion of group elements is fast: this is true for example for elliptic curves, groups of rational divisor classes of hyperelliptic curves, trace zero varieties and XTR. The methods can also be used for computing single exponentiations in groups which admit an appropriate automorphism satisfying a monic equation of small degree over the integers.