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The equivalence may be read on the regular continued fraction representation, as shown by the following theorem of Serret: Theorem: Two irrational numbers x and y are equivalent if and only if there exist two positive integers h and k such that the regular continued fraction representations of x and y
A similar problem, involving equating like terms rather than coefficients of like terms, arises if we wish to de-nest the nested radicals + to obtain an equivalent expression not involving a square root of an expression itself involving a square root, we can postulate the existence of rational parameters d, e such that
The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions.
Indeterminate form is a mathematical expression that can obtain any value depending on circumstances. In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function.
Algebraic simplification; for example, the CAS can combine multiple terms into one fraction by finding a common denominator. Evaluation of trigonometric expressions to exact values. For example, sin(60°) returns 3 2 {\displaystyle {\frac {\sqrt {3}}{2}}} instead of 0.86603.
The difference of two squares can also be illustrated geometrically as the difference of two square areas in a plane.In the diagram, the shaded part represents the difference between the areas of the two squares, i.e. .