A simple and accurate method to calculate square roots and cube roots for Chemistry students without calculator.

Presented here are two simple formulae, for the calculation of square and cube roots that are frequently encountered in Chemistry courses, at school and college levels (whether undergraduate or graduate). However, tests are dicult to be designed which do not provide for a calculator and which require the calculation of these quantities. The formulae are simple interval-weighted denominator method based to get an accurate value of these quantities. This will enable students to quickly and accurately compute square roots and cube roots. Apart from of


Introduction
Square roots and cube roots appear frequently in many chemical equations. A simple interval-weighted denominator method to get an accurate value of these quantities is therefore, desirable at all levels of Chemistry programs, whether undergraduate or graduate. The method described in this paper is a convenient one that can be used in an exam setting easily, without a use of a calculator. Alternatively, quicker results can be obtained with the use of a non-programmable calculator which lacks the cube root function. The method described can be adapted to school and college Chemistry curricula with equal ease. Apart from Chemistry, this formula can well be adapted in all other elds which require the use of square roots and cube roots of numbers.

Methodology
A. Determination of Square Roots (SR): In this method, a knowledge of the perfect squares between which the desired number lies must be known. Between 1 and 1000, there are 31 perfect squares, namely, 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,576,625,676,729,784,841, 900 and 961. In this method: a) Determine the numbers which are perfect squares between which the given number say, X) is present. In this method, a knowledge of the perfect cubes between which the cube root of the desired number (X) lies must be known. Thus, the interval within which the number X lies is important. As an example, let us take the determination of cube roots of numbers from 1 to 1000. Beyond 1000, the numbers can be expressed in the form p × 10 q , where p is any number in between 1 to 1000 whose cube root can be determined easily by this method and q is any natural number. In this method: is determined for each interval separately and is calculated using eq. (iii) given in Table 1.
In function form, the quantity B can be written as Where B is the oor function in E and E = smaller perfect cube of the interval within which X lies. This function is, however, applicable for only eight intervals from 8-27 to 729-1000. For the interval 1-8, the ceiling function may be used.

Results And Discussion
(i) Square Roots The square roots of a few selected numbers from 1 to 100 calculated by eq. (i) are shown in Table 2 along with the actual square roots. The square roots of the earlier numbers, 2 and 3 present in the rst interval of 1-4 are not as accurate compared to the actual values. Accuracies improve from 5, where they reach 99.9% and improve to 99.99% beyond 25, an excellent result. In numbers beyond 100, accuracies of 99.999% are achieved, considered a near perfect result in a practical setting. Such accuracies obtained from this method are acceptable in a real life chemistry setting, whether at school or college. Accurate square roots of 2 and 3 can be obtained by using the square roots of 200 and 300 and dividing the result by 10, a result that gives excellent values. In cases where the obtained and actual square root values are the same, yet percentage accuracy is not 100% because the gures have been rounded off to three decimal places whereas the accuracies have been calculated on seven decimal places, a practice usually followed in Chemistry courses around the world, where rounding off is usually done at the end.

(ii) Cube Roots
A comparison of the values of cube roots obtained from this method for numbers in all the nine intervals from 1-8 to 729-1000 are shown in Table 3. In each interval, a number at the beginning, one in the middle and one at the end have been taken. In this table, it can be observed that even for smaller numbers in the rst interval, the determination of cube root of the number near the end of the interval is fairly high (99.9% or more). The accuracy of the middle number is also fairly high in the rst interval itself and is acceptable but increases tremendously (99.9% or more) from the 64-125 interval onwards. On the whole, these formulae can be easily adapted to chemistry curricula, both at the school and college (undergraduate and graduate) levels and may act to supplement other methods in promoting calculatorfree tests.

Conclusions
In conclusion, simple, accurate and adaptable formulae have been described to calculate square roots and cube roots of numbers. These formulae can be used universally in a simple school or college setting in a Chemistry course, whether General or Physical Chemistry courses wherein tests are administered without the provision of calculators for the test-takers. The various parameters described in the two formulae are easy to memorize and give accurate and quick results.

Declarations
Con ict of Interest: