Square Root of -54

The square root of a negative number involves the imaginary unit 'i,' where i² = -1. Therefore, the square root of -54 can be expressed as √(-54) = √(54) × i. The square root of 54 itself is expressed in both radical and exponential forms. In radical form, it is √54, and in exponential form, it is (54)^(1/2). Since 54 is not a perfect square, √54 = 7.348469, which is an irrational number. Thus, the square root of -54 is 7.348469i.

Finding the square root of a negative number like -54 involves understanding imaginary numbers. Since traditional methods like prime factorization, long division, and approximation apply only to positive numbers, we instead express the result in terms of 'i'. Let us explore the following approach:

• Imaginary number approach

Imaginary numbers are used to find the square roots of negative numbers. Here’s how you can express √(-54):

Step 1: Express -54 as -1 × 54.

Step 2: The square root of -1 is 'i', so √(-54) = √(54) × √(-1) = √54 × i.

Step 3: Simplify √54. The prime factorization of 54 is 2 × 3 × 3 × 3, which can be expressed as √(2 × 3^3).

Step 4: Simplify further to get 3√6.

Step 5: Therefore, the square root of -54 is 3√6 × i.

Students commonly make errors while dealing with square roots of negative numbers. Understanding the role of imaginary numbers is crucial. Let’s discuss typical mistakes and how to avoid them.

• Imaginary number: A number that can be written as a real number multiplied by the imaginary unit 'i', where i² = -1.
   
• Complex number: A number comprising a real and an imaginary part, expressed as a + bi, where a and b are real numbers.
   
• Radical: An expression that includes a square root, cube root, or higher root.
   
• Prime factorization: The expression of a number as a product of its prime factors.
   
• Irrational number: A number that cannot be expressed as a simple fraction or ratio of two integers.