113 LearnersLast updated on May 26th, 2025
Square Root of -224
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.
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.
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
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.
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.
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.
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
Ignoring the Imaginary Unit
It is essential to remember that the square root of a negative number involves the imaginary unit, i.
For example, √-9 = 3i, not 3.
Mistake 2
Confusing Real and Imaginary Roots
Ensure students understand the difference between real and imaginary roots.
For example, the square root of -1 is i, not a real number.
Mistake 3
Misapplying Square Root Methods
Applying methods suited for positive numbers to negative numbers can lead to errors. Remember to multiply the square root of the positive part by i.
Mistake 4
Misunderstanding the Concept of i
Students often confuse the imaginary unit, i, with integers or other variables. Reinforce that i is defined as √-1.
Mistake 5
Mistakes in Prime Factorization
When factoring, ensure that students correctly determine the prime factors of the positive component before multiplying by i.
Square Root of -224 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √-144?
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.
Problem 2
A square-shaped building measures -224 square feet; what is the length of each side?
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.
Problem 3
Calculate 5 times the square root of -224.
Approximately 74.833i
Explanation
First, find the square root of -224, which is 14.9666i.
Then multiply by 5: 5 × 14.9666i = 74.833i.
Problem 4
What will be the square root of (-144 + 6)?
Approximately 12.083i
Explanation
First, find the sum (-144 + 6) = -138.
The square root of -138 is approximately √138 * i = 11.7473i.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √-49 units, and the width ‘w’ is 10 units.
The perimeter is 20 + 14i units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Length = √-49 = 7i.
Perimeter = 2 × (7i + 10) = 20 + 14i units.
FAQ on Square Root of -224
1.What is √-224 in its simplest form?
2.What is the imaginary unit i?
3.How do you find the square root of a negative number?
4.Is the square root of a negative number real?
5.What are complex numbers?
6.How does learning Algebra help students in Qatar make better decisions in daily life?
7.How can cultural or local activities in Qatar support learning Algebra topics such as Square Root of -224?
8.How do technology and digital tools in Qatar support learning Algebra and Square Root of -224?
9.Does learning Algebra support future career opportunities for students in Qatar?
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 Qatar
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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
