In this paper, the authors proposed an approximate hybrid square rooter (AHSQR). It is the combination of array and logarithmic-based square rooter (SQR) to create a balance between accuracy and hardware performance. An array-based SQR is utilized as an exact SQR (ESQR) to obtain the MSBs of output for high precision, while a logarithmic SQR is used to estimate the remaining output digits to enhance design metrics. A modified AHSQR (MAHSQR) is also proposed to retain accuracy at increasing degrees of approximation by computing the square root of LSBs using the ESQR unit. This reduces the mean relative error distance by up to 31% and the normalized mean error distance by up to 26%. Various accuracy metrics and hardware characteristics are evaluated and analyzed for 16-bit unsigned exact, state-of-the-art, and proposed SQRs. The proposed SQRs are designed using Verilog and implemented using Artix7 FPGA. The results show that the proposed SQRs performances are improved compared to the state-of-the-art methods by being approximately 70% smaller, 2.5 times faster, and consuming only 25% of the power of the ESQR. Applications of the proposed SQRs as a Sobel edge detector, and K-means clustering for image processing, and an envelope detector for communication systems are also included.