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Last updated on May 26th, 2025

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Square Root of -400

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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.

Square Root of -400 for UAE Students
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What is 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).

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Finding the Square Root of -400

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.

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Square Root of -400 by Prime Factorization Method

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.

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Square Root of -400 by Long Division Method

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.

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Square Root of -400 by Approximation Method

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.

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Common Mistakes and How to Avoid Them in the Square Root of -400

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.

Mistake 1

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Forgetting the Imaginary Unit 'i'

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It is crucial to remember that the square root of a negative number involves the imaginary unit 'i'.

For instance, the square root of -400 is 20i, not just 20.

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Square Root of -400 Examples

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Problem 1

Can you help Alex determine the result of multiplying √(-400) by 3?

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The result is 60i.

Explanation

First, find the square root of -400, which is 20i.

Then, multiply by 3: 20i × 3 = 60i.

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Problem 2

A complex number is given as 5 + √(-400). What is its form?

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The complex number is 5 + 20i.

Explanation

The square root of -400 is 20i, so the complex number becomes 5 + 20i.

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Problem 3

Calculate 2 × √(-400) - 10i.

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The result is 30i.

Explanation

First, calculate 2 × √(-400): 2 × 20i = 40i.

Then, subtract 10i: 40i - 10i = 30i.

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Problem 4

What is the square of √(-400)?

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The square is -400.

Explanation

The square root of -400 is 20i.

Squaring it gives (20i)^2 = 400 × i^2 = 400 × (-1) = -400.

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Problem 5

Find the product of √(-400) and its complex conjugate.

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The result is 400.

Explanation

The conjugate of 20i is -20i.

The product is 20i × (-20i) = -400i^2 = 400.

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FAQ on Square Root of -400

1.What is the square root of -400 in its simplest form?

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2.What are the factors of 400?

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3.Calculate the square of 20i.

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4.Is -400 a perfect square?

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5.What is the imaginary unit 'i'?

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6.How does learning Algebra help students in United Arab Emirates make better decisions in daily life?

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7.How can cultural or local activities in United Arab Emirates support learning Algebra topics such as Square Root of -400?

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8.How do technology and digital tools in United Arab Emirates support learning Algebra and Square Root of -400?

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9.Does learning Algebra support future career opportunities for students in United Arab Emirates?

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Important Glossaries for the Square Root of -400

  • Square root: The operation that finds a number which, when multiplied by itself, gives the original number. For negative numbers, this involves imaginary numbers.

 

  • Imaginary unit: The imaginary unit 'i' is defined as √(-1). It is used to express the square roots of negative numbers.

 

  • Complex number: A number composed of a real part and an imaginary part. Example: 5 + 20i.

 

  • Conjugate: For a complex number a + bi, the conjugate is a - bi. For example, the conjugate of 3 + 4i is 3 - 4i.

 

  • Perfect square: A number that is the square of an integer. Negative numbers cannot be perfect squares.
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About BrightChamps in United Arab Emirates

At BrightChamps, we know algebra is more than symbols—it opens doors to endless possibilities! We’re dedicated to helping kids throughout the UAE develop crucial math skills, focusing today on the Square Root of -400 with a special focus on square roots—in an enjoyable, engaging, and easy-to-understand way. Whether your child is figuring out the speed of a roller coaster at Dubai Parks and Resorts, keeping track of local football scores, or budgeting their allowance for the latest gadgets, mastering algebra gives them confidence for everyday situations. Our interactive lessons make learning both fun and simple. Because kids in the UAE learn in varied ways, we customize our teaching to fit each child’s needs. From Dubai’s soaring skyscrapers to Abu Dhabi’s cultural heritage, BrightChamps brings math alive, making it relevant and exciting throughout the UAE. Let’s make square roots a joyful part of every child’s math journey!
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Jaskaran Singh Saluja

About the Author

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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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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