Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$(\sqrt{-10})^{2}$$

Step-by-Step Solution

Answer

The result in standard form after performing the operation is -10.

Step 1: Rewrite the expression using imaginary unit

The square root of -10 can be written as \(i\sqrt{10}\) because the square root of a negative number involves the imaginary unit 'i'.

Step 2: Square the expression

When squaring any quantity, multiply the quantity by itself. In this case, squaring \(i\sqrt{10}\) results in \( (i\sqrt{10})^2 = i^2 \times 10\).

Step 3: Simplify the expression

Since \(i^2\) is equal to -1, the expression simplifies to \( -1 \times 10 = -10\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit is fundamental in understanding complex numbers. The imaginary unit, denoted as 'i', has one peculiar property that sets it apart: it is defined as the square root of -1. Normally, square roots are used for finding a number that, when multiplied by itself, gives a positive product. So, the concept of a square root of a negative number doesn't fit into the real number system. That's where 'i' comes into play.
By definition, we have that:
\[ i^2 = -1 \].
The incorporation of the imaginary unit expands the real number system to the complex number system, where numbers can have both real and imaginary parts. In the exercise involving \(\sqrt{-10}\), 'i' is used to express the square root as \(i\sqrt{10}\), because you cannot have a real number as a square root of a negative number.

Squaring Complex Numbers

When squaring complex numbers, the operation is similar to squaring real numbers, but with an extra step to accommodate the imaginary unit. To square a complex number, you multiply it by itself. However, with the presence of 'i', we need to remember the definition of the imaginary unit and apply it.
In our example, when squaring \(i\sqrt{10}\), you get \((i\sqrt{10})^2\), which distributes to \(i^2 \times 10\).

Applying the Imaginary Unit

Remembering that \(i^2 = -1\), we replace the \(i^2\) in the expression with -1, simplifying our squared complex number accordingly. This step is crucial because it effectively takes our expression back into the realm of real numbers, which are often easier to work with and more intuitive to understand.

Standard Form of a Complex Number

The standard form of a complex number is a way to systematically express these numbers. It is written as \(a + bi\), where 'a' is the real part and 'bi' is the imaginary part.
In this form, 'a' and 'b' are real numbers while 'i' is the imaginary unit. The standard form allows for straightforward addition, subtraction, multiplication, and division of complex numbers by treating 'i' as an algebraic term.
Returning to the exercise, once we square \(i\sqrt{10}\) and use the property of the imaginary unit, we conclude with the real number -10. While this specific example resulted in a real number, most squared complex numbers stay in complex form, but always remember to look out for and simplify terms involving \(i^2\), due to its definition. By doing so, you ensure the number is in its simplest standard form, making it easy to use in further calculations.