Consider the following numbers: A. 22.5^ B. 22.5^ C. 22.5^ - 22.5^ How many of the above are irrational numbers?

Step-by-step solution

Let t = 22.5^ = 45^ 2 Now, 45^ = (2 45 2 )^ = 2 ( 45 2 )^ 1- ^2 ( 45 2 )^ aligned & 1= 2 t 1-t^2 t^2+2 t-1=0 & t= -2 4+4 2 =-1 2 & ( 45 2 )^ =-1 2 ( 45 2 )^ =-1+( 2 ) aligned . In I ^ st quadrant is positive (II) Now, 22.5^ = 1 22.5^ = 1 -1+ 2 =1+ 2 which is also an irrational number. (III) Since, 22.5^ - 22.5^ =-1+ 2 -(1+ 2 )=-2 which is not an irrational number.