Question
Here is the question : GRADE 11: WHAT IS THE SQUARE ROOT OF -64?
Option
Here is the option for the question :
- 8
- 16
- -8
- 8i
The Answer:
And, the answer for the the question is :
Explanation:
Since no integer multiplied by itself can equal -64, as eleventh graders learn, the answer contains the letter “i,” which stands for “imaginary number.”
In Grade 11 mathematics, students explore complex numbers and their properties. Complex numbers extend the concept of real numbers by introducing the imaginary unit (i), which is defined as the square root of -1. Complex numbers play a crucial role in various fields of mathematics, physics, and engineering. Let’s delve into an example involving the square root of -64. By understanding the properties of complex numbers and applying the rules of square roots, we find that the square root of -64 is 8i.
The square root of a number is a value that, when multiplied by itself, gives the original number. In the case of real numbers, the square root of a positive number always has a real value. However, when dealing with negative numbers, the square root does not have a real value and requires the use of complex numbers.
To find the square root of -64, we start by expressing -64 as a product of a positive number and -1:
-64 = 64 × -1.
Next, we can simplify the expression by taking the square root of both sides:
(\sqrt{-64} = \sqrt{64 \times -1}).
Using the property (\sqrt{ab} = \sqrt{a} \times \sqrt{b}), we can split the square root of the product into the product of the square roots:
(\sqrt{-64} = \sqrt{64} \times \sqrt{-1}).
The square root of 64 is 8, and the square root of -1 is defined as the imaginary unit (i). Therefore, we can rewrite the equation as:
(\sqrt{-64} = 8i).
Hence, the square root of -64 is determined to be 8i.
Understanding complex numbers and their properties is essential as they provide a powerful mathematical framework for solving problems that involve both real and imaginary quantities. Grade 11 students who develop a solid understanding of complex numbers lay the foundation for more advanced topics such as complex analysis, electrical engineering, and quantum mechanics.
Complex numbers find applications in various fields, including electrical engineering, signal processing, and physics. They are used to represent and analyze alternating currents, harmonic oscillations, and quantum states, among other phenomena. By mastering the concept of complex numbers, students gain the ability to model and solve real-world problems that involve both real and imaginary components.
Furthermore, understanding complex numbers enhances students’ mathematical and problem-solving skills. It enables them to manipulate and analyze mathematical expressions involving complex quantities, solve equations involving complex numbers, and visualize mathematical relationships in the complex plane. By studying complex numbers, students also develop a deeper appreciation for the interconnectedness of different areas of mathematics and the elegance of mathematical structures.
complex numbers offer a rich set of operations and properties, including addition, subtraction, multiplication, and division. These operations allow for the manipulation and simplification of complex expressions, enabling students to solve complex equations and analyze complex systems.
in Grade 11 mathematics, understanding complex numbers and their properties is a fundamental skill. By applying the rules of square roots and recognizing the imaginary unit (i), we can find the square root of a negative number. In the case of -64, the square root is determined to be 8i. Mastering the concepts of complex numbers not only improves mathematical proficiency but also enhances students’ problem-solving abilities and opens doors to a wide range of mathematical applications in various fields.