Last updated on May 26th, 2025
The square of a number is obtained by multiplying the number by itself. The inverse operation is finding the square root. The square root is applicable in various fields like engineering and physics. Here, we will explore the square root of -148.
The square root is the inverse operation of squaring a number. Since -148 is a negative number, its square root is not a real number. Instead, it is expressed in terms of an imaginary number. The square root of -148 can be written as √(-148) = √148 × √(-1) = 12.1655i, where 'i' is the imaginary unit satisfying i² = -1.
Imaginary numbers arise when we take the square root of negative numbers. In standard form, an imaginary number is denoted as 'bi', where 'b' is a real number and 'i' is the imaginary unit. Understanding imaginary numbers involves recognizing that they extend the real number system to solve equations that do not have real solutions, such as x² = -1.
To find the square root of the positive part of -148, which is 148, we proceed with standard methods used for real numbers:
Step 1: Express 148 in terms of its prime factors: 2 x 2 x 37.
Step 2: Simplify the square root of 148: √148 = √(2² × 37) = 2√37.
Step 3: Calculate the approximate value: 2√37 ≈ 12.1655. Hence, the square root of -148 is 12.1655i.
We can approximate the square root of 148, a part of -148, using nearby perfect squares:
Step 1: Identify the perfect squares closest to 148: 144 (12²) and 169 (13²).
Step 2: Since 148 is closer to 144, start with an approximation close to 12. Further refine using methods like interpolation or calculator to obtain √148 ≈ 12.1655.
Remember, the square root of -148 is then 12.1655i.
Imaginary numbers, including those like the square root of -148, are used in advanced fields such as electrical engineering, quantum physics, and complex number theory. They help in analyzing circuits, describing wave functions, and solving differential equations where real numbers are insufficient.
Mistakes often occur when dealing with square roots of negative numbers. It's crucial to acknowledge the role of imaginary units. Here's a look at common errors and how to prevent them.
Can you help Alex find the area of a square with side length √(-148)?
The area cannot be determined as a real number since the side length is imaginary.
The side length √(-148) = 12.1655i is imaginary.
Since area must be a real number, a square with imaginary side length does not have a real area.
Find the product of √(-148) × 3.
36.4965i
First, find √(-148) = 12.1655i.
Then multiply by 3: 12.1655i × 3 = 36.4965i.
What is the square root of (-148)²?
148
When squaring a square root, the result is the absolute value of the original number: √((-148)²) = |148| = 148.
Calculate √(-148) × √(-1).
-12.1655
√(-148) = 12.1655i and √(-1) = i, so multiplying them gives (12.1655i) × i = 12.1655 × i² = -12.1655.
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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