175 LearnersLast updated on August 5th, 2025
Square Root of -90
When a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The concept of square roots extends into complex numbers when dealing with negative values. Here, we will discuss the square root of -90.
What is the Square Root of -90?
The square root is the inverse of squaring a number. Since -90 is a negative number, its square root is not a real number. Instead, it is a complex number. The square root of -90 is expressed in both radical and exponential form. In radical form, it is expressed as √(-90), whereas in exponential form, it is (-90)^(1/2). The square root of -90 is an imaginary number, which can be expressed as 3√10i, where i is the imaginary unit defined as √(-1).
Finding the Square Root of -90
The square root of a negative number involves the imaginary unit i. For positive numbers, methods like prime factorization, long division, and approximation are used. However, for negative numbers, we focus on expressing them in terms of i. Let us explore this concept further: Imaginary unit i Expressing negative roots in terms of i Simplifying under the radical
Square Root of -90 by Expressing in Terms of i
To express the square root of a negative number using imaginary numbers, we follow these steps:
Step 1: Express -90 as -1 × 90.
Step 2: Recognize that the square root of -1 is i, the imaginary unit. Thus, √(-1) = i.
Step 3: Write √(-90) as √(-1 × 90) = √(-1) × √90 = i√90.
Step 4: Simplify √90 by finding its prime factors: 90 = 2 × 3^2 × 5. Thus, √90 = √(3^2 × 10) = 3√10.
Step 5: Combine the expressions to get √(-90) = 3√10i.
Understanding Complex Numbers in the Context of Square Roots
Complex numbers, which include real and imaginary parts, are essential for understanding square roots of negative numbers. The square root of -90 is purely imaginary, expressed as 3√10i. Real numbers cannot yield a negative square when squared; hence, imaginary numbers are used. Let's elaborate on this: Real part: zero for purely imaginary numbers Imaginary part: derived from the radical simplification Application of complex numbers in various fields
Applications and Importance of Imaginary Numbers
Imaginary numbers extend the real number system to solve equations lacking real solutions. Their applications include electrical engineering, quantum physics, and complex number theory. Understanding square roots of negative numbers is foundational in these fields: Electrical circuits: alternating current representations Quantum mechanics: wave functions and probabilities Complex analysis: advanced calculus and engineering
Common Mistakes and How to Avoid Them with the Square Root of -90
Mistakes frequently occur when dealing with square roots of negative numbers due to misunderstandings about imaginary units and complex numbers. Let us explore common errors and how to avoid them:
Mistake 1
Confusing Real and Imaginary Parts
It's crucial to distinguish between real and imaginary components of complex numbers.
For instance, 3√10i is purely imaginary with no real part, unlike a number like 3 + 3√10i, which has both.
Mistake 2
Misapplying Real Number Methods
Using methods like prime factorization meant for real numbers on negative numbers leads to errors. Only the concept of i applies to negative square roots.
For example, √(-90) should not be simplified using real number techniques.
Mistake 3
Neglecting the Imaginary Unit
Forgetting to incorporate the imaginary unit i in square roots of negative numbers results in incorrect answers. Always include i when dealing with square roots of negatives, such as √(-90) = 3√10i.
Mistake 4
Incorrect Simplification of Radicals
Proper simplification of radicals is essential. Incorrect simplification of √90, such as ignoring the prime factors, leads to errors. Correctly simplify √90 to 3√10 before multiplying by i.
Mistake 5
Misunderstanding of Complex Number Operations
Complex numbers involve both real and imaginary parts. Misunderstanding operations with these parts can cause mistakes, such as assuming √(-90) is a real number or misapplying complex arithmetic rules.
Square Root of -90 Examples
Problem 1
Can you help Max find the magnitude of a complex number if it is given as 5 + √(-90)?
The magnitude of the complex number is approximately 20.12.
Explanation
The magnitude of a complex number a + bi is √(a^2 + b^2). For 5 + √(-90), express √(-90) as 3√10i. Thus, we have 5 + 3√10i. Calculate: √(5^2 + (3√10)^2) = √(25 + 90) = √115 ≈ 20.12.
Problem 2
If the square root of -90 is added to a real number 10, what is the result in complex number form?
The result is 10 + 3√10i.
Explanation
The square root of -90 is expressed as 3√10i. Adding this to the real number 10 gives the complex number 10 + 3√10i, with 10 as the real part and 3√10i as the imaginary part.
Problem 3
Calculate the result of squaring the square root of -90.
The result is -90.
Explanation
Squaring the square root of -90, which is 3√10i, results in (3√10i)^2 = (3√10)^2 × i^2 = 90 × (-1) = -90, since i^2 = -1.
Problem 4
What is the conjugate of the complex number 7 + √(-90)?
The conjugate is 7 - 3√10i.
Explanation
The conjugate of a complex number a + bi is a - bi. For 7 + √(-90), express √(-90) as 3√10i. Thus, the conjugate is 7 - 3√10i.
Problem 5
Find the imaginary part of the complex number formed by adding 5 to the square root of -90.
The imaginary part is 3√10i.
Explanation
Form the complex number by adding 5 to the square root of -90, which is 3√10i. Therefore, the complex number is 5 + 3√10i, with the imaginary part being 3√10i.
FAQ on Square Root of -90
1.What is √(-90) in its simplest form?
2.What are the applications of imaginary numbers like √(-90)?
3.Why can't the square root of -90 be a real number?
4.Can the square root of a negative number be simplified without using i?
5.What is the significance of the imaginary unit i?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Square Root of -90?
8.How do technology and digital tools in India support learning Algebra and Square Root of -90?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of -90
- Square root: The inverse operation of squaring a number. For negative numbers, it involves imaginary numbers. Example: The square root of -90 is 3√10i.
- Imaginary number: A number in the form of a real number multiplied by the imaginary unit i, representing the square root of negative values. Example: 3i is an imaginary number.
- Complex number: A number comprising a real part and an imaginary part, expressed as a + bi. Example: 5 + 3√10i is a complex number.
- Imaginary unit (i): A mathematical constant defined as √(-1), enabling calculations with negative square roots.
- Conjugate: For a complex number a + bi, the conjugate is a - bi, important in complex arithmetic. Example: The conjugate of 7 + 3√10i is 7 - 3√10i.
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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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