If p^3=q^4=r^6=t^7=s^2 , then _t(p q r s)= .

Step-by-step solution

Let p^3 = q^4 = r^6 = t^7 = s^2 = k . Then p = k^ 1/3 , q = k^ 1/4 , r = k^ 1/6 , t = k^ 1/7 , and s = k^ 1/2 . The product pqrs = k^ 1/3 k^ 1/4 k^ 1/6 k^ 1/2 = k^ (1/3) + (1/4) + (1/6) + (1/2) . Converting to a common denominator of 12 yields 4 12 + 3 12 + 2 12 + 6 12 = 15 12 = 5 4 , so pqrs = k^ 5/4 . Evaluating _t(pqrs) gives _ k^ 1/7 (k^ 5/4 ) = 5/4 1/7 _k k = 5 4 7 = 35 4 . Final answer: 35 4 .