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The number dd is irrational. Which statement about 24d\sqrt{24} - d is true?\newlineChoices:\newline(A) 24d\sqrt{24} - d is rational.\newline(B) 24d\sqrt{24} - d is irrational.\newline(C) 24d\sqrt{24} - d can be rational or irrational, depending on the value of dd.

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Q. The number dd is irrational. Which statement about 24d\sqrt{24} - d is true?\newlineChoices:\newline(A) 24d\sqrt{24} - d is rational.\newline(B) 24d\sqrt{24} - d is irrational.\newline(C) 24d\sqrt{24} - d can be rational or irrational, depending on the value of dd.
  1. Nature of 24\sqrt{24}: First, let's consider the nature of 24\sqrt{24}. 24\sqrt{24} can be simplified to 4×6\sqrt{4\times6}, which is 2×62\times\sqrt{6}. Since 6\sqrt{6} is not a perfect square, it is an irrational number. Therefore, 24\sqrt{24} is irrational.
  2. Subtraction of Irrational Numbers: Now, let's consider the subtraction of two numbers: an irrational number 24\sqrt{24} and another irrational number dd. The difference between two irrational numbers can be either rational or irrational. However, there is no general rule that guarantees the result will be rational or irrational without knowing the specific values of the two irrational numbers.
  3. Indeterminate Result: Since we do not have the specific value of dd, we cannot determine the exact nature of 24d\sqrt{24} - d. Therefore, without additional information about dd, we cannot conclude whether 24d\sqrt{24} - d is rational or irrational.
  4. Given Choices: Given the choices provided:\newline(A) 24d\sqrt{24} - d is rational.\newline(B) 24d\sqrt{24} - d is irrational.\newline(C) 24d\sqrt{24} - d can be rational or irrational, depending on the value of dd.\newlineThe correct choice is (C) because the nature of the expression depends on the specific value of the irrational number dd.

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