Square Root of -90

The square root is the inverse of squaring a number. Since -90 is a negative number, its square root is not a real number. Instead, it is a complex number. The square root of -90 is expressed in both radical and exponential form. In radical form, it is expressed as √(-90), whereas in exponential form, it is (-90)^(1/2). The square root of -90 is an imaginary number, which can be expressed as 3√10i, where i is the imaginary unit defined as √(-1).

The square root of a negative number involves the imaginary unit i. For positive numbers, methods like prime factorization, long division, and approximation are used. However, for negative numbers, we focus on expressing them in terms of i. Let us explore this concept further: Imaginary unit i Expressing negative roots in terms of i Simplifying under the radical

To express the square root of a negative number using imaginary numbers, we follow these steps:

Step 1: Express -90 as -1 × 90.

Step 2: Recognize that the square root of -1 is i, the imaginary unit. Thus, √(-1) = i.

Step 3: Write √(-90) as √(-1 × 90) = √(-1) × √90 = i√90.

Step 4: Simplify √90 by finding its prime factors: 90 = 2 × 3^2 × 5. Thus, √90 = √(3^2 × 10) = 3√10.

Step 5: Combine the expressions to get √(-90) = 3√10i.

Complex numbers, which include real and imaginary parts, are essential for understanding square roots of negative numbers. The square root of -90 is purely imaginary, expressed as 3√10i. Real numbers cannot yield a negative square when squared; hence, imaginary numbers are used. Let's elaborate on this: Real part: zero for purely imaginary numbers Imaginary part: derived from the radical simplification Application of complex numbers in various fields

Imaginary numbers extend the real number system to solve equations lacking real solutions. Their applications include electrical engineering, quantum physics, and complex number theory. Understanding square roots of negative numbers is foundational in these fields: Electrical circuits: alternating current representations Quantum mechanics: wave functions and probabilities Complex analysis: advanced calculus and engineering

• Square root: The inverse operation of squaring a number. For negative numbers, it involves imaginary numbers. Example: The square root of -90 is 3√10i.
   
• Imaginary number: A number in the form of a real number multiplied by the imaginary unit i, representing the square root of negative values. Example: 3i is an imaginary number.
   
• Complex number: A number comprising a real part and an imaginary part, expressed as a + bi. Example: 5 + 3√10i is a complex number.
   
• Imaginary unit (i): A mathematical constant defined as √(-1), enabling calculations with negative square roots.
   
• Conjugate: For a complex number a + bi, the conjugate is a - bi, important in complex arithmetic. Example: The conjugate of 7 + 3√10i is 7 - 3√10i.