Last updated on May 26th, 2025
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.
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 \( \sqrt{-243} = 15.588i \).
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
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 \(i \times \sqrt{243}\).
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: \( \sqrt{-243} = 9 \times \sqrt{3} \times i \)
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: \( \sqrt{-243} = 15.588i \)
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.
Can you help Max find the imaginary part of the square root of -243?
The imaginary part is 15.588i.
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.
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 \(0 + 15.588i\).
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 \(0 + 15.588i\).
Calculate \(5 \times \sqrt{-243}\).
The result is \(77.94i\).
First, find the square root of -243, which is \(15.588i\).
Then, multiply by 5: \(5 \times 15.588i = 77.94i\).
What will be the square root of \(-729\)?
The square root is \(27i\).
Find the square root of the positive part, 729, which is 27.
The square root of -729 involves imaginary unit \(i\), so it is \(27i\).
Find the imaginary part of the square root of -81.
The imaginary part is \(9i\).
The square root of 81 is 9.
Thus, the square root of -81 is \(9i\), with an imaginary part of 9.
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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