Last updated on May 26th, 2025
The square of a number is obtained when it is multiplied by itself. The inverse of this operation is the square root. Square roots have applications in various fields, such as engineering and physics. In this discussion, we will explore the square root of -400.
The square root is the inverse of squaring a number. Since -400 is a negative number, its square root involves the use of imaginary numbers. The square root of -400 can be expressed in terms of the imaginary unit 'i'. In radical form, it is written as √(-400), and in exponential form, it is expressed as (-400)^(1/2). The square root of -400 is 20i, where 'i' represents the imaginary unit √(-1).
For negative numbers, the square root involves imaginary numbers. The commonly used methods for finding square roots of positive numbers do not directly apply to negative numbers. Instead, we use the concept of imaginary numbers. Here are the basic steps:
1. Factor out -1 from the number.
2. Find the square root of the positive part.
3. Combine it with 'i', the imaginary unit.
Although prime factorization is useful for finding square roots of positive integers, it does not directly apply to negative numbers. However, we can still determine the square root of -400 using its positive factors:
Step 1: Factor 400 into its prime factors: 2 x 2 x 2 x 2 x 5 x 5, which is 2^4 x 5^2.
Step 2: The square root of 400 is the product of half the power of each factor: (2^2) x 5 = 20.
Step 3: Combine this with 'i', the imaginary unit, to account for the negative sign: 20i.
The long division method is typically used for finding square roots of positive numbers. Since -400 is negative, this method does not directly apply. Instead, we recognize that the square root of -400 involves the imaginary unit 'i': 1. Consider the square root of the positive part, √400 = 20. 2. Combine this with the imaginary unit to obtain the square root of -400: 20i.
The approximation method is generally used to estimate square roots of non-perfect squares. For -400, we directly use the understanding of imaginary numbers:
Step 1: Consider the square root of the positive part, √400 = 20.
Step 2: Combine with 'i' to get the square root of -400: 20i.
Students often make errors when dealing with the square roots of negative numbers, such as ignoring the imaginary unit or misapplying methods for positive numbers. Let's look at some common mistakes and how to avoid them.
Can you help Alex determine the result of multiplying √(-400) by 3?
The result is 60i.
First, find the square root of -400, which is 20i.
Then, multiply by 3: 20i × 3 = 60i.
A complex number is given as 5 + √(-400). What is its form?
The complex number is 5 + 20i.
The square root of -400 is 20i, so the complex number becomes 5 + 20i.
Calculate 2 × √(-400) - 10i.
The result is 30i.
First, calculate 2 × √(-400): 2 × 20i = 40i.
Then, subtract 10i: 40i - 10i = 30i.
What is the square of √(-400)?
The square is -400.
The square root of -400 is 20i.
Squaring it gives (20i)^2 = 400 × i^2 = 400 × (-1) = -400.
Find the product of √(-400) and its complex conjugate.
The result is 400.
The conjugate of 20i is -20i.
The product is 20i × (-20i) = -400i^2 = 400.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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