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 root extends into the field of complex numbers when dealing with negative numbers. Here, we will discuss the square root of -71.
The square root is the inverse of the square of the number. Since -71 is a negative number, its square root cannot be expressed as a real number. Instead, it is expressed as a complex number. The square root of -71 is expressed in the form of \(i\sqrt{71}\), where \(i\) is the imaginary unit. The number \(\sqrt{71}\) is approximately 8.42615, so the square root of -71 is approximately \(8.42615i\).
The concept of square roots of negative numbers is rooted in complex numbers. When dealing with negative numbers under a square root, the imaginary unit \(i\) is used, where \(i^2 = -1\). The square root of a negative number is not defined in the set of real numbers. Let us now go through the steps to understand this:
To express the square root of -71 in terms of complex numbers, we use the imaginary unit \(i\).
Step 1: Identify the negative number inside the square root, which is -71.
Step 2: Express \(\sqrt{-71}\) as \(\sqrt{71} \times \sqrt{-1}\).
Step 3: Simplify to \(i\sqrt{71}\).
Thus, the square root of -71 is \(i\sqrt{71}\), which approximately equals \(8.42615i\).
Although -71 does not have a real square root, we can approximate the square root of 71, which is used when calculating the expression \(i\sqrt{71}\).
1. Identify the perfect squares around 71, which are 64 and 81.
2. Recognize that \(\sqrt{64} = 8\) and \(\sqrt{81} = 9\).
3. Approximate \(\sqrt{71}\) as a value between 8 and 9, closer to 8.5.
Using a calculator, \(\sqrt{71}\) is approximately 8.42615, so \(\sqrt{-71} \approx 8.42615i\).
When working with square roots of negative numbers, students often make errors. Let’s discuss some common mistakes and how to avoid them:
Do not forget to include the imaginary unit \(i\) when dealing with the square roots of negative numbers.
For example, \(\sqrt{-25} = 5i\), not \(5\).
Students can make mistakes when finding the square root of negative numbers, often due to misunderstanding the role of the imaginary unit. Let's explore these in detail.
Can you help Max express the square root of -49 in terms of the imaginary unit?
The square root of -49 is \(7i\).
The square root of -49 is expressed as \(\sqrt{49} \times \sqrt{-1} = 7i\).
If the expression \((\sqrt{-71})^2\) is simplified, what is the result?
The result is -71.
When you square the square root of a number, you get the original number.
Thus, \((\sqrt{-71})^2 = -71\).
Calculate \(3 \times \sqrt{-71}\).
The result is \(25.27845i\).
First, find \(\sqrt{-71} = 8.42615i\).
Then multiply by 3: \(3 \times 8.42615i = 25.27845i\).
What is the square root of \(-100\)?
The square root is \(10i\).
\(\sqrt{-100} = \sqrt{100} \times \sqrt{-1} = 10i\).
Express \(\sqrt{-144}\) in terms of \(i\).
The expression is \(12i\).
The square root of -144 is \(\sqrt{144} \times \sqrt{-1} = 12i\).
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.
: He loves to play the quiz with kids through algebra to make kids love it.