MATHEMATICS 23 



sary, because it is impossible to mistake the operation that 

 has to be performed. But, since 7 + -^-= * -, , if 



175 ^ 66 is substituted for 7+^, the heavy line becomes 



necessary in order to make the resulting expression clear. 

 Thus, 



160 160 160 



660 175 + 660 835 

 + 25 25 25 



Fractional exponents are sometimes used instead of the 

 radical sign; that is, instead of indicating the square, cube, 

 fourth root, etc. of some quantity, as 37, by V37, ^37~, ^3T, 

 etc., those roots are indicated 37^, 37$, 37^, etc. Should 

 the numerator of the fractional exponent be some quantity 

 other than 1, this quantity, whatever it may be, indicates 

 that the quantity affected by the exponent is to be raised to 

 the power indicated by the numerator; the denominator is 

 always the index of the root. Hence, instead of expressing 

 the cube root of the square of 37 as "V37 2 , it may be expressed 

 37 3, the denominator being the index of the root; in other 

 words, ^372 = 37*. Likewise, ^(l + a 2 &)3 may also be writ- 

 ten (l+a 2 *))^, a much simpler expression. 



Several examples showing how to apply some of the more 

 difficult formulas will now be given. 



The area of any segment of a circle that is less than (or 

 equal to) a semicircle is expressed by the formula 

 , irr z E c , . . 



^TW-Jfr-*- 



in which A =area of segment; ir = 3.1416; r radius; E = angle 

 obtained by drawing lines from the center to the extremities 

 of arc of segment; c = chord of segment; and h = height of 

 segment. 



EXAMPLE. What is the area of a segment whose chord is 

 10 in. long, angle subtended by chord is 83.46, radius is 

 7.5 in., and height of segment is 1.91 in.? 



SOLUTION. Applying the formula just given, 

 . nr^E c. .. 3.1416X7.52X83.46 10 ,_ _ 



A = -36Q-2 (r ~ K > 360 2-C7.5-1.91) 



= 40.968-27.95 = 13.018 sq. in., nearly 



