What is the square root of negative 289
The Correct Answer and Explanation is:
The square root of negative 289 is an imaginary number. To express this, we need to involve the imaginary unit iii, where i=−1i = \sqrt{-1}i=−1.
To solve for the square root of -289, we can break it down as follows: −289=(−1)×289\sqrt{-289} = \sqrt{(-1) \times 289}−289=(−1)×289
Using the property of square roots, which states that the square root of a product is the product of the square roots, we can separate the negative sign: −289=−1×289\sqrt{-289} = \sqrt{-1} \times \sqrt{289}−289=−1×289
Now, we know that −1=i\sqrt{-1} = i−1=i and 289=17\sqrt{289} = 17289=17. Therefore: −289=i×17=17i\sqrt{-289} = i \times 17 = 17i−289=i×17=17i
Thus, the square root of negative 289 is 17i17i17i.
Explanation:
The square root of a negative number cannot be a real number, as no real number multiplied by itself gives a negative result. This is where the concept of imaginary numbers comes into play. Imaginary numbers allow us to extend the number system beyond just real numbers. The unit iii is defined specifically to handle the square roots of negative numbers.
In this case, iii is used to indicate the square root of -1, and through it, we can represent the square root of any negative number as a multiple of iii. The number 17i17i17i is purely imaginary because it has no real component (no “real part”). This is a critical extension in mathematics, especially in fields like complex analysis and electrical engineering, where both real and imaginary numbers are used together to describe phenomena that can’t be captured by real numbers alone.
Thus, the square root of -289 is 17i17i17i.
