Last updated on May 26th, 2025
The square root is the inverse of the square of a number. When dealing with negative numbers, square roots become complex numbers. Here, we will discuss the square root of -162.
The square root of a negative number involves complex numbers. The square root of -162 is expressed in both radical and exponential form involving the imaginary unit i. In radical form, it is expressed as √-162 = √162 × i, whereas in exponential form as (162)^(1/2) × i. The value of √162 is approximately 12.7279, making the square root of -162 approximately 12.7279i, which is a complex number.
To find the square root of a negative number, we need to use the concept of imaginary numbers. Let's explore the methods used to calculate the square root of -162:
1. Split into real and imaginary parts
2. Simplify the square root of the positive part
3. Combine with the imaginary unit i
To find the square root of -162, we follow these steps:
Step 1: Recognize that -162 can be rewritten as (-1) × 162.
Step 2: The square root of -162 becomes √(-1) × √162.
Step 3: The square root of -1 is i (the imaginary unit).
Step 4: Calculate √162, which is approximately 12.7279.
Step 5: Multiply √162 by i to get the square root of -162 as approximately 12.7279i.
The approximation method for finding the square roots of a positive number can be applied here to find the real part of the complex square root.
Step 1: Identify the closest perfect squares around 162. They are 144 (12^2) and 169 (13^2).
Step 2: Use linear interpolation to estimate √162. (162 - 144) / (169 - 144) = 0.72
Step 3: Calculate 12 + 0.72 = 12.72
Thus, √162 is approximately 12.7279, and the square root of -162 is 12.7279i.
When dealing with negative square roots, it's easy to make mistakes by not accounting for the imaginary unit or by misapplying methods suitable for real numbers. Let's look at some common errors and how to prevent them.
Can you find the square root of -98 and explain it?
The square root is approximately 9.899i.
To find the square root of -98, split it as √(-1) × √98.
The square root of -1 is i, and √98 is approximately 9.899.
Thus, the square root of -98 is 9.899i.
If the length of a side of a square is √-50, what is the area of the square?
The area is -50 square units.
The area of the square = side^2.
If the side length is √-50, the area = (√-50) * (√-50) = -50 square units.
Calculate √-200 × 3.
Approximately 42.4264i.
First, find the square root of -200, which is approximately 14.1421i.
Then, multiply by 3: 14.1421i × 3 = 42.4264i.
What is the square root of (-64 - 36)?
The square root is 10i.
First, find the sum of (-64 - 36), which is -100.
The square root of -100 = √(-1) × √100 = 10i.
If a rectangle's dimensions are √-162 by 10, what is its area?
The area is -1620 square units.
The area of a rectangle = length × width.
With dimensions √-162 and 10, the area = (√-162) * 10 = -1620 square units.
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.