Question
Explain why √−9 is not possible but 3√−27 is possible. Use complete sentences!
Answers
GPT-4o mini
The expression √−9 is not possible in the realm of real numbers because the square root of a negative number does not have a real solution; it requires the concept of imaginary numbers. Specifically, the square root of -1 is denoted as 'i,' which is the imaginary unit. Therefore, √−9 can be expressed as √9 * √−1 = 3i, indicating that it results in an imaginary number rather than a real number.
On the other hand, the expression 3√−27 is possible because it involves a cube root instead of a square root. The cube root of a negative number is defined in the realm of real numbers. In this case, the cube root of -27 is -3, since (-3) * (-3) * (-3) = -27. Therefore, 3√−27 simplifies to 3 * (-3) = -9, which is a real number. Thus, while √−9 is not possible in real numbers, 3√−27 is valid, resulting in a real number.
On the other hand, the expression 3√−27 is possible because it involves a cube root instead of a square root. The cube root of a negative number is defined in the realm of real numbers. In this case, the cube root of -27 is -3, since (-3) * (-3) * (-3) = -27. Therefore, 3√−27 simplifies to 3 * (-3) = -9, which is a real number. Thus, while √−9 is not possible in real numbers, 3√−27 is valid, resulting in a real number.