Final Answer:
The number that can be multiplied by any irrational number to obtain an irrational number is (Option c) 16^√16.
Step-by-step explanation:
To determine which number can be multiplied by any irrational number to yield an irrational result, let's analyze the options. We are looking for a number with an exponent that involves an irrational number(Option c).
Option c) 16^√16 fits this criterion. To understand why, consider that √16 equals 4. Therefore, 16^√16 is equivalent to 16^4. This involves raising 16 to the power of 4, which is a rational number, but the process itself involves an irrational exponent (√16), making the result irrational.
In contrast, options a), b), and d) all have rational exponents or involve square roots of perfect squares, resulting in rational numbers. For example, option a) 2√2 simplifies to 2 × √2 = 2√2, which is rational. Thus, the only option that meets the criteria of being multiplied by any irrational number to yield an irrational result is c) 16^√16.
In conclusion, the choice of c) 16^√16 stands out as the answer, as it involves an irrational exponent, making it a suitable candidate for multiplication by any irrational number to produce an irrational outcome.