In this study, the effect of correlated and uncorrelated power law noise on the random deposition and random deposition with surface relaxation models is investigated. In simple forms of these models with white noise, the particles randomly select a site for deposition; however, in this study, these models are simulated with a correlated power noise where the distance between deposition sites follows a power law relation as δx = int [r ^ (- 1 / (۲ρ))], and with uncorrelated power noise where the particles size is determined via the relation P (l) ~ l ^ (- (μ + 1)). In the above relations, r is a random number with a uniform distribution in the interval (0, 1), ρ is the correlation intensity and µ is the noise exponent. The results show that the scaling of these models is completely different and multi-fractal features are observed in the power law noises. The growth exponent for the random deposition model with various values ​​of correlation intensity ρ is a constant value equal to β = 0.5 ± 0.02, whereas for the random deposition with surface relaxation model, this exponent enhances from β = 0.25 ± 0.02 in ρ = 0 (equivalent to simple version of this model with white noise) to β = 0.5 ± 0.02 in ρ = 1 and remains constant. According to the results, uncorrelated power law noise leads to a step function enhancement of roughness width at different growth times. The fractal analysis performed by the Multi-fractal detrended fluctuation analysis(MFDFA) method illustrates the multi-fractal behavior of the simulated rough surfaces.