Math IOQM

7. If \( x^{(x^4)} = 4 \), what is the value of \( x^{(x^2)} + x^{(x^8)} \)? Answer: 258 Solution 1: Guesswork x^4 . log(x) = 2.log(2) = 4.log(sqrt(2)) = (sqrt 2)^4 . log(sqrt(2)) So x = sqrt(2) Solution 2: x x 4 = 4 x^{x^4} = 4 . Let us replace 4 of the exponent of left hand side by x x 4 x^{x^4} then equation becomes x x x 4 = 4 x^{x^{x^4}} = 4 . Repeating this process infinitely we get x x x x ⋯ = 4 x^{x^{x^{x^{\cdots}}}} = 4 . Now we replace the exponent of left hand side by 4 and equation now becomes x 4 = 4 x^4 = 4 hence x = 2 x = \sqrt{2} or x = − 2 x = -\sqrt{2} . So x x 2 + x x 8 = 258 x^{x^2} + x^{x^8} = 258 Solution 3: Given  x x 4 = 4 x^{x^4} = 4 Now raise to the power 4 on both sides. ( x x 4 ) 4 = 4 4 \left(x^{x^4}\right)^4 = 4^4 We know that ( x m ) n = x m n = ( x n ) m (x^m)^n = x^{mn} = (x^n)^m ... (1) In LHS, m = x 4 m = x^4 and n = 4 n = 4 Hence LHS = ( x 4 ) ( x 4 ) (x^4)^{(x^4)} (using the first and third terms of the equality in (1)) Let y = x 4 y = ...

Suppose that \( 4^{x_{1}} \) = 5, \( 5^{x_{2}} \) = 6 \( \ldots \)and \( 127^{x_{124}} \) = 128. What is the value of the product X1.X2....X124 ? Answer: 7/2.

A country has a food deficit of 10 % . Its population grows continuously at a rate of 3 % per year. Its annual food production every year is 4% more than that of the last year. Assuming that the average food requirement per person remains constant, prove that the country will become self-sufficient in food after n years, where n is the smallest integer bigger than or equal to (ln 10 - ln 9)/ (ln ( 1.04 ) - 0.03). Solution: Key point to note here is that population is growing continuously whereas food production is growing annually. So P_n = P_0.e^(rt) F_n = F_0.(1 + r/100)^t

Answer: 1