20 9 3 = p^2 t^7 > > In the given equation, p 1 and t 1. If 1 t = p^3n^-, where n is a constant, what is the value of n?
","rationale_html":"41 The correct answer is 81. An expression of the form n a m , where m and n are $ m integers greater than 1 and a 0, is equivalent to an. Therefore, the expression 2 2 9 3 3= 7 on the left-hand side of the given equation is equivalent to p ; thus, p t . If = 3 n -1 3 n -1 t p , where n is a constant, then substituting p for t in the equation yields p 3 2 = p 7 9 ^ 3 n -1 h . It’s given that p > 1; therefore, 3 2 = 7 9 ^ 3 n -1 h. Multiplying each 21 14=27 3 -1 side of this equation by yields ^ n h. Distributing multiplication over 14=81 -27 27 subtraction yields n . Adding to each side of this equation yields 41 41=81 81 = n. Dividing each side of this equation by yields 81 n. Therefore, if t = p 3 n -1 , where n is a constant, the value of n is 8 4 1 1 . Note that 41/81, .5061, .5062, and 0.506 are examples of ways to enter a correct answer.
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