Constructible numbers are the numbers that can be constructed by using compass and straightedge in a finite sequence. They can be produced from natural numbers using only addition, subtraction, multiplication, division, and square root operations. These operations can be repeated, which creates more complicated expressions for a mathematical object. Calculation by computers only gives an approximation of the exact value, which could lead to a loss of accuracy. An alternative to approximation is exact arithmetic, which is the computation to find an exact value without rounding errors. In this thesis, we have presented a method of computing with the exact value of constructible numbers, specifically the rational numbers and its field extension as well as repeated field extension, through the basic operations. However, we only limit our implementation to the quadratic polynomial and the operations between two numbers of the same extension field. Future work on polynomials with higher degrees and algorithms to include operations with numbers from different extension fields and expression of number as an element of a new extension field remains to be done.