116 LearnersLast updated on May 26th, 2025
Square Root of -243
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The concept of square roots extends to complex numbers when dealing with negative numbers. Here, we will discuss the square root of -243.
What is the Square Root of -243?
The square root is the inverse of the square of the number. Since -243 is a negative number, its square root is expressed using complex numbers. In the radical form, it is expressed as \( \sqrt{-243} \), whereas in exponential form, it is \((-243)^{1/2}\). Since the square root of a negative number involves imaginary numbers, \( \sqrt{-243} = i \times \sqrt{243} \). Here, \( \sqrt{243} = 15.588 \), so \( \sqrt{-243} = 15.588i \).
Finding the Square Root of -243
The square root of a negative number is found using imaginary numbers. We can use the following methods:
1. Expressing using imaginary unit \(i\)
2. Simplifying the square root of the positive part
3. Combining to form a complex number
Square Root of -243 by Expressing Using Imaginary Unit \(i\)
To find the square root of -243, we first express the number as a product of -1 and 243: -243 = -1 × 243
Step 1: The square root of -1 is \(i\), where \(i\) is the imaginary unit.
Step 2: The square root of 243 is calculated separately.
Step 3: Combine the results to express the square root of -243 as \(i \times \sqrt{243}\).
Square Root of -243 by Simplifying the Positive Part
To simplify the square root of the positive part, 243, use prime factorization:
Step 1: Find the prime factors of 243: 243 = 3 × 3 × 3 × 3 × 3 = \(3^5\)
Step 2: Simplify using the property of square roots: \( \sqrt{243} = \sqrt{3^5} = 3^2 \times \sqrt{3} = 9 \times \sqrt{3} \)
Step 3: Combine with the imaginary unit: \( \sqrt{-243} = 9 \times \sqrt{3} \times i \)
Square Root of -243 by Forming a Complex Number
By forming a complex number, the square root of a negative number is expressed as follows:
Step 1: Recognize that the square root of any negative number can be expressed with \(i\).
Step 2: Calculate the square root of the positive number 243, which is approximately 15.588.
Step 3: Express the result as a complex number: \( \sqrt{-243} = 15.588i \)
Common Mistakes and How to Avoid Them in the Square Root of -243
Students often make errors when calculating the square root of negative numbers, such as ignoring the imaginary unit \(i\) or misunderstanding the concept of imaginary numbers. Let us look at a few common mistakes in detail.
Mistake 1
Ignoring the Imaginary Unit \(i\)
It is crucial to remember that the square root of a negative number involves the imaginary unit \(i\). Forgetting to include \(i\) leads to incorrect results.
For example, \(\sqrt{-4}\) is \(2i\), not 2.
Mistake 2
Incorrect Simplification of Square Roots
Students may incorrectly simplify square roots by not properly factoring the number. Always ensure correct prime factorization is used.
For example, \(\sqrt{50} \neq 5\); the correct form is \(\sqrt{50} = 5\sqrt{2}\).
Mistake 3
Confusing Real and Imaginary Numbers
It is important to differentiate between real and imaginary numbers. Only real numbers have non-imaginary square roots. Be cautious when dealing with square roots of negative numbers.
Mistake 4
Misusing Mathematical Symbols
Misusing the square root symbol or forgetting it altogether can lead to errors. Ensure the square root is clearly marked when performing calculations.
Mistake 5
Errors in Prime Factorization
Errors in the prime factorization can lead to incorrect simplifications. Double-check the factors to avoid mistakes in calculations.
Square Root of -243 Examples
Problem 1
Can you help Max find the imaginary part of the square root of -243?
The imaginary part is 15.588i.
Explanation
The square root of -243 is expressed as \(i \times \sqrt{243}\).
The square root of 243 is approximately 15.588.
So, the imaginary part is 15.588i.
Problem 2
If a complex number is given as \(a + bi\), where \(a = 0\) and \(|b| = \sqrt{243}\), what is the complex number?
The complex number is \(0 + 15.588i\).
Explanation
Given that \(|b|\) is the absolute value of the imaginary part, and it equals \(\sqrt{243}\), which is 15.588.
Thus, the complex number is \(0 + 15.588i\).
Problem 3
Calculate \(5 \times \sqrt{-243}\).
The result is \(77.94i\).
Explanation
First, find the square root of -243, which is \(15.588i\).
Then, multiply by 5: \(5 \times 15.588i = 77.94i\).
Problem 4
What will be the square root of \(-729\)?
The square root is \(27i\).
Explanation
Find the square root of the positive part, 729, which is 27.
The square root of -729 involves imaginary unit \(i\), so it is \(27i\).
Problem 5
Find the imaginary part of the square root of -81.
The imaginary part is \(9i\).
Explanation
The square root of 81 is 9.
Thus, the square root of -81 is \(9i\), with an imaginary part of 9.
FAQ on Square Root of -243
1.What is \(\sqrt{-243}\) in its simplest form?
2.What are the prime factors of 243?
3.Calculate the square of 243.
4.Is -243 a prime number?
5.Why is the square root of a negative number imaginary?
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 -243?
8.How do technology and digital tools in India support learning Algebra and Square Root of -243?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of -243
- Imaginary number: An imaginary number is a number that can be written as a real number multiplied by the imaginary unit \(i\), where \(i^2 = -1\).
- Complex number: A complex number is a number that has both a real and an imaginary part, written in the form \(a + bi\).
- Prime factorization: The process of expressing a number as the product of its prime factors.
- 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 an imaginary unit.
- Complex plane: A plane used to represent complex numbers graphically, with the real part on the x-axis and the imaginary part on the y-axis.
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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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