This section is from the book "The Building Trades Pocketbook", by International Correspondence Schools. Also available from Amazon: Building Trades Pocketbook: a Handy Manual of reference on Building Construction.
Involution is the process of multiplying a number by itself one or more times, the product obtained being called a certain power of the number. If the number is multiplied by itself, the result is called the square of the number; thus, 9 is the square of 3, since 3X3 = 9. If the square of a number is multiplied by the number, the result is called the cube of the number; thus, 27 is the cube of 3, since 3X3X3 = 27. The power to which a number is to be raised is indicated by a small figure, called an exponent, placed to the right and a little above the number; thus, 72 means that 7 is to be squared; 273 means that 27 is to be cubed, etc.
The operations of involution present no difficulty, as nothing but multiplication is involved, the number of times the number is to be taken as a factor being shown by the exponent.
If the number is a fraction, raise both numerator and denominator to the power indicated.
A valuable little rule to memorize for finding the square of a mixed number in which the fraction is 1/2, as 3 1/2, 10 1/2, etc., is as follows: Multiply the next less whole number by the next greater, and add 1/4. For example, the square of 6 1/2 is 6 (the next less number) x 7 (the next greater)+1/4= 42 1/4; (19 1/2)2 = 19X20 + 1/4 = 380 1/4; (8 1/2)2 = 8X9 + 1/4= 72 1/4; etc.