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The relative escape rate (RER) in the bistable sawtooth system driven by correlated white noises was studied in the case of linear multiplicative noise coefficient.And we compare the RER of linear multiplicative noise coefficient with the RER of the piecewise constant multiplicative noise coefficient in the bistable sawtooth system,finally compare the RER of the bistable sawtooth potential with the RER of the quartic potential in the case of linear multiplicative noise coefficient.It can be seen that the form of the multiplicative noise coefficient and the nonlinearity of potential have an important influence on the resonant activation and the suppression of the RER.The resonant activation of the RER vanishes in the negative correlation when the multiplicative noise coefficient was shifted from piecewise constant to linear function in the bistable sawtooth potential.The suppression of the RER disappears in the negative correlation when thelinear potential is converted into nonlinear one.``