Question 21

The two quantities are equal.

Both Quantity A and Quantity B simplify to the same value when evaluated. Specifically, Quantity A simplifies to \((1/100)^N\) and Quantity B simplifies to \((1/10) \cdot 1^N = (1/10) \cdot 1 = (1/10)\), leading to equality when multiplied appropriately.

A) Quantity A is greater.

This option is incorrect because Quantity A, which is \((-1/10)^{2N}\), results in a positive value of \((1/100)^N\). Depending on the value of \(N\), it does not exceed Quantity B in all instances.

B) Quantity B is greater.

This option is also incorrect. Quantity B, which evaluates to \((1/10)(-1)^{2N}\), resolves to \((1/10) \cdot 1 = (1/10)\), which is not greater than Quantity A when both are evaluated at the same \(N\).

C) The two quantities are equal.

This option is correct as both quantities simplify to the same expression. The evaluation of \((-1/10)^{2N}\) yields a positive result equal to \((1/100)^N\), which is equivalent to the outcome of Quantity B when simplified.

D) The relationship cannot be determined from the information given.

This option is incorrect. Since both quantities can be evaluated and simplified to show equality, it is clear that their relationship is determinable and specifically that they are equal.

Conclusion

The correct answer is that the two quantities are equal, as both simplify to the same positive expression. Other options fail because they either misinterpret the simplification of the quantities or suggest uncertainty where none exists. Thus, the equality is definitive for all positive integer values of \(N\).