Simplify the the square root of -121
The Correct Answer and Explanation is:
The square root of -121 is an imaginary number because the square root of any negative number does not exist in the real number system. Let’s break this down step-by-step.
Step 1: Understanding the square root of a negative number
The square root of a number ( x ) is defined as a number that, when multiplied by itself, gives ( x ). In other words, ( \sqrt{x} = y ) if ( y \times y = x ).
However, for any negative number ( x ), there is no real number ( y ) such that ( y \times y = x ) because the product of two real numbers is always non-negative. For example:
- The square root of ( 9 ) is ( 3 ) because ( 3 \times 3 = 9 ).
- But there is no real number that, when squared, results in a negative number like ( -9 ), since ( (-3) \times (-3) = 9 ), which is positive.
Step 2: Imaginary numbers
To address the square root of negative numbers, mathematicians use the concept of imaginary numbers. The imaginary unit, denoted as ( i ), is defined as the square root of ( -1 ), i.e.,
[
i = \sqrt{-1}
]
With this, we can now express the square root of negative numbers. Specifically, the square root of ( -121 ) can be written as:
[
\sqrt{-121} = \sqrt{121} \times \sqrt{-1} = 11i
]
Step 3: Conclusion
Thus, the square root of ( -121 ) is ( 11i ), where ( i ) is the imaginary unit. This result belongs to the complex number system, where numbers are composed of both real and imaginary parts. Imaginary numbers are crucial in various fields, including engineering, physics, and mathematics, particularly when dealing with waveforms, electrical circuits, and complex equations.
So, the square root of ( -121 ) simplifies to:
[
\boxed{11i}
]