114 LearnersLast updated on May 26th, 2025
Square Root of -245
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 vehicle design, finance, etc. Here, we will discuss the square root of -245.
What is the Square Root of -245?
The square root is the inverse of the square of a number. Since -245 is a negative number, its square root is not a real number. The square root of -245 is expressed in terms of imaginary numbers. In radical form, it is expressed as √(-245) = √245 * i, where i is the imaginary unit (i = √-1). In the exponential form, it is expressed as (245)^(1/2) * i. √245 ≈ 15.6525, so the square root of -245 is approximately 15.6525i, which is an imaginary number.
Finding the Square Root of -245
For negative numbers, the square root involves imaginary numbers. Here, we are looking for a number that, when squared, gives -245. This number is represented using the imaginary unit i. Let us consider the following methods used for calculating the square roots of positive numbers and modify them for our purpose:
- Prime factorization method
- Long division method
- Approximation method
Square Root of -245 by Prime Factorization Method
The prime factorization method involves expressing a number as a product of its prime factors. Let's see how we can express 245:
Step 1: Finding the prime factors of 245 Breaking it down, we get 5 × 7 × 7 = 5 × 7².
Step 2: With these prime factors, the square root of 245 in terms of real numbers is √245 = √(5 × 7²) = 7√5.
Since -245 is negative, we include the imaginary unit: √(-245) = 7√5 * i.
Square Root of -245 by Long Division Method
The long division method is typically used for non-perfect square numbers. However, since -245 is negative, we have to consider the imaginary unit.
Step 1: Calculate the square root of the positive part, which is 245, using long division to get approximately 15.6525.
Step 2: Since the original number is negative, the square root of -245 is 15.6525i.
Square Root of -245 by Approximation Method
The approximation method involves finding the square root of the positive part and then multiplying by the imaginary unit.
Step 1: Identify the closest perfect squares around 245, which are 225 (15²) and 256 (16²). So √245 is between 15 and 16.
Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). For approximation: (245 - 225) / (256 - 225) = 20/31 ≈ 0.645.
Thus, √245 ≈ 15 + 0.645 = 15.645 and therefore, √(-245) ≈ 15.645i.
Common Mistakes and How to Avoid Them in the Square Root of -245
Students often make mistakes while finding the square root, especially with negative numbers where imaginary numbers come into play. Let’s look at some common mistakes and how to avoid them.
Mistake 1
Forgetting about the imaginary unit
It's crucial to remember that the square root of a negative number involves the imaginary unit i.
For example: √(-50) = √50 * i = 7.071i, both ±7.071i should be considered.
Mistake 2
Not using the square root symbol properly
Some errors occur by not correctly applying the square root symbol. It's important to simplify internally before taking the root.
For instance, √(12 + 24) = √(36) = 6, not √12 + √24.
Mistake 3
Ignoring the negative sign in front of the number
When dealing with negative numbers, don’t ignore the sign. Remember to denote the result using the imaginary unit i.
For example, √(-50) = √50 * i = 7.071i.
Mistake 4
Confusing square root with cube root
It’s common to mix up square root and cube root symbols. Ensure you know the difference, such as √(-50) and ∛(-50) are different, with the former involving imaginary numbers.
Mistake 5
Errors in approximation
When approximating square roots, be careful with the decimal places and ensure calculations are precise. Approximation requires careful attention to detail to avoid errors.
Square Root of -245 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √(-138)?
The area cannot be determined as a real number because the side length is imaginary.
Explanation
The area of a square = side².
Since the side length is √(-138), it involves imaginary numbers, and squaring it results in a negative area, which is not meaningful in real terms.
Problem 2
A square-shaped building measuring -245 square feet is claimed to exist; what can you infer about this claim?
The claim is invalid in real-world terms.
Explanation
Area cannot be negative in reality.
The mention of -245 square feet likely involves an error or a misunderstanding, as square footage is always a positive measure.
Problem 3
Calculate √(-245) × 5.
Approximately 78.2625i
Explanation
First, calculate √(-245) ≈ 15.6525i.
Then, multiply by 5: 15.6525i × 5 = 78.2625i.
Problem 4
What will be the square root of (-138 + 6)?
The square root is approximately 11.4455i
Explanation
First, calculate the expression: -138 + 6 = -132.
Then, √(-132) = √132 * i ≈ 11.4455i.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √(-138) units and the width ‘w’ is 38 units.
The perimeter cannot be determined as a real number.
Explanation
Perimeter = 2 × (length + width). With length √(-138), the calculation involves imaginary numbers, making the perimeter not meaningful in real terms.
FAQ on Square Root of -245
1.What is √(-245) in its simplest form?
2.Mention the factors of 245.
3.Calculate the square of -245.
4.Is -245 a prime number?
5.Is the square root of a negative number real?
6.How does learning Algebra help students in Australia make better decisions in daily life?
7.How can cultural or local activities in Australia support learning Algebra topics such as Square Root of -245?
8.How do technology and digital tools in Australia support learning Algebra and Square Root of -245?
9.Does learning Algebra support future career opportunities for students in Australia?
Important Glossaries for the Square Root of -245
- 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: Denoted as i, it is defined as the square root of -1.
- Imaginary number: A number that can be expressed in the form of a real number multiplied by i.
- Prime factorization: Breaking down a number into a product of its prime factors.
- Approximation: Estimating a value when an exact figure is not necessary or possible.
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Jaskaran Singh Saluja
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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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