answer:Let A be the number of apples. According to the problem, for some integers d, e, and f, we have: A = 9d + 2, A = 10e + 2, A = 11f + 2. This implies that A - 2 is divisible by 9, 10, and 11. We need to find the LCM of these numbers: - The LCM of 9 and 10 is 90 (since 9 = 3^2 and 10 = 2 cdot 5). - The LCM of 90 and 11 (which is prime) is 990. Thus, 990 divides A - 2. The smallest possible value of A greater than 2 is A = 990 + 2 = boxed{992}.

answer:The value (x = f^{-1}(-7/64)) corresponds to the solution of (f(x) = -7/64). Thus, we set up the equation [ frac{x^7 - 1}{4} = -frac{7}{64}. ] Multiplying through by 4 gives [ x^7 - 1 = -frac{7}{16}. ] Adding 1 to each side results in [ x^7 = -frac{7}{16} + frac{16}{16} = frac{9}{16}. ] Solving for (x), we find [ x = sqrt[7]{frac{9}{16}}. ] The exact value of (x) is found by taking the seventh root: [ x = left(frac{9}{16}right)^{frac{1}{7}}. ] Thus, the inverse function evaluated at (-7/64) is [ f^{-1}left(-frac{7}{64}right) = boxed{left(frac{9}{16}right)^{frac{1}{7}}}. ]

answer:To solve for the range of the independent variable x in the function y=frac{x}{5-x}, we must ensure the denominator is not equal to zero to avoid division by zero. Thus, we set up the inequality: 1. Start with the condition for the denominator: 5-x neq 0. 2. Solve for x: begin{align*} 5 - x &neq 0 5 &neq x x &neq 5. end{align*} This means that the value of x cannot be 5 for the function to be defined. Therefore, the range of the independent variable x is all real numbers except 5. Thus, the correct answer is boxed{D}.

answer:1. Convert ( frac{2}{7} ) and ( frac{1}{6} ) to have a common denominator of 84: [ frac{2}{7} = frac{24}{84}, quad frac{1}{6} = frac{14}{84}. ] This gives the range ( frac{14}{84} < frac{N}{84} < frac{24}{84} ). 2. To find the integer values of ( N ) that satisfy this inequality, we solve: [ 14 < N < 24. ] The integer values of ( N ) in this range are 15, 16, 17, 18, 19, 20, 21, 22, 23. 3. The average of the smallest and largest values (15 and 23) is: [ text{Average} = frac{15 + 23}{2} = frac{38}{2} = 19. ] Therefore, the average of all integer values of ( N ) is ( boxed{19} ).