Digital signal processing (DSP) is fundamentally dependent on multiplication for tasks like image processing, audio processing, communication systems, and control systems. Because multiplication significantly impacts processing speed, faster multipliers are essential for boosting overall system performance. This proj ect investigates the implementation and utilization of a Look-Up Table (LUT) multiplier on a Field Programmable Gate Array (FPGA) to address this challenge. The proposed design capitalizes on the inherent capabilities of the FPGA architecture to create optimized and efficient multipliers. We will showcase an 8-bit multiplier constructed entirely from basic LUT components. In addition to this exploration, the proj ect delves into the development of approximate multipliers. These multipliers prioritize lower power consumption and design simplicity, while maintaining an acceptable level of accuracy. To achieve this balance, specific selection rules are employed to identify potential zero coefficients within the data. The efficacy of the multiplier design was rigorously evaluated through a three step process: development in Verilog HDL, simulation, and synthesis within a Vivado Zynq FPGA. This evaluation assessed critical factors such as area usage on the FPGA chip, processing delay, and overall power consumption. Finally, to demonstrate the computational efficiency of the approximate multiplier in real-world scenarios, it was integrated into the structure of an FIR filter.