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Last updated on May 26th, 2025

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Square Root of -224

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If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in various fields, including mathematics and engineering. Here, we will discuss the square root of -224, a complex number.

Square Root of -224 for Filipino Students
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What is the Square Root of -224?

The square root is the inverse of the square of the number. Since -224 is a negative number, its square root involves an imaginary unit, i. The square root of -224 is expressed in both radical and exponential form. In radical form, it is expressed as √-224, whereas in exponential form, it is (-224)^(1/2). To express it in terms of real and imaginary numbers, we write √-224 = √224 * i, where √224 ≈ 14.9666. Hence, the square root of -224 is approximately 14.9666i, indicating that it is an imaginary number.

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Finding the Square Root of -224

The prime factorization method is used for perfect square numbers. However, the square root of negative numbers involves imaginary numbers. Therefore, we use the multiplication of the square root of the positive part with the imaginary unit, i. Let's explore the following methods:

 

  • Prime factorization method (for positive components)
  • Imaginary unit multiplication
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Square Root of -224 by Prime Factorization Method

The prime factorization of a number involves expressing it as a product of prime numbers. Now let us look at how 224 is broken down into its prime factors:

 

Step 1: Finding the prime factors of 224 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 7: 2^5 x 7

 

Step 2: Now, we found the prime factors of 224. Since -224 is negative, we use the imaginary unit, i.

 

Thus, √-224 = √224 * i = 2^2.5 * √7 * i.

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Square Root of -224 by Imaginary Unit Multiplication

The imaginary unit multiplication method is used to find the square root of negative numbers. Let us now learn how to find the square root of -224 using this method:

 

Step 1: First, find the square root of the positive part of -224.

 

Step 2: The square root of 224 is approximately 14.9666.

 

Step 3: Multiply the result by the imaginary unit, i, to get √-224 = 14.9666i.

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Square Root of -224 by Approximation Method

The approximation method can also be extended to include imaginary numbers when dealing with negative square roots. Let us learn how to approximate the square root of -224:

 

Step 1: Determine the closest perfect squares to 224. The closest are 196 (14^2) and 225 (15^2), so √224 is between 14 and 15.

 

Step 2: Approximate √224 as 14.9666 and multiply by i to find the square root of -224: 14.9666i.

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Common Mistakes and How to Avoid Them in the Square Root of -224

Students often make mistakes while finding the square root of negative numbers, such as not accounting for the imaginary unit or applying methods suitable only for positive numbers. Let's explore some common mistakes:

Mistake 1

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Ignoring the Imaginary Unit

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

For example, √-9 = 3i, not 3.

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Square Root of -224 Examples

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Problem 1

Can you help Max find the area of a square box if its side length is given as √-144?

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The area of the square is -144 square units.

Explanation

The area of the square = side^2.

The side length is √-144 = 12i.

Area of the square = (12i)^2 = -144.

Therefore, the area of the square box is -144 square units.

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Problem 2

A square-shaped building measures -224 square feet; what is the length of each side?

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Approximately 14.9666i feet.

Explanation

The side length of a square is the square root of its area.

So, the side length is √-224 = 14.9666i feet.

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Problem 3

Calculate 5 times the square root of -224.

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Approximately 74.833i

Explanation

First, find the square root of -224, which is 14.9666i.

Then multiply by 5: 5 × 14.9666i = 74.833i.

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Problem 4

What will be the square root of (-144 + 6)?

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Approximately 12.083i

Explanation

First, find the sum (-144 + 6) = -138.

The square root of -138 is approximately √138 * i = 11.7473i.

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Problem 5

Find the perimeter of a rectangle if its length ‘l’ is √-49 units, and the width ‘w’ is 10 units.

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The perimeter is 20 + 14i units.

Explanation

Perimeter of the rectangle = 2 × (length + width).

Length = √-49 = 7i.

Perimeter = 2 × (7i + 10) = 20 + 14i units.

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FAQ on Square Root of -224

1.What is √-224 in its simplest form?

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2.What is the imaginary unit i?

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3.How do you find the square root of a negative number?

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4.Is the square root of a negative number real?

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5.What are complex numbers?

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6.How does learning Algebra help students in Philippines make better decisions in daily life?

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7.How can cultural or local activities in Philippines support learning Algebra topics such as Square Root of -224?

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8.How do technology and digital tools in Philippines support learning Algebra and Square Root of -224?

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9.Does learning Algebra support future career opportunities for students in Philippines?

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Important Glossaries for the Square Root of -224

  • Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For negative numbers, it involves the imaginary unit, i.

 

  • Imaginary unit (i): The imaginary unit i is defined as the square root of -1, allowing the representation of square roots of negative numbers.

 

  • Complex number: A number composed of a real and an imaginary component, expressed as a + bi.

 

  • Prime factorization: The process of breaking down a number into its smallest prime factors.

 

  • Approximation: Estimating a value close to the actual value, often used in finding the square roots of non-perfect squares.
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About BrightChamps in Philippines

At BrightChamps, we believe algebra is more than symbols—it’s a gateway to countless opportunities! Our mission is to support children across the Philippines in mastering key math skills, focusing today on the Square Root of -224 with an emphasis on understanding square roots—in a lively, fun, and easy-to-follow way. Whether your child is calculating the speed of a roller coaster at Enchanted Kingdom, tracking basketball scores, or managing their allowance to buy the latest gadgets, mastering algebra builds their confidence for daily challenges. Our hands-on lessons make learning simple and enjoyable. Since kids in the Philippines learn in different styles, we tailor our approach to each learner. From Manila’s busy streets to the beautiful islands of Palawan, BrightChamps brings math to life, making it exciting and relevant throughout the Philippines. Let’s make square roots a fun 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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Max, the Girl Character from BrightChamps

Fun Fact

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

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