Problem 8 - Introductory Algebra for College Students

Question

Express each number in terms of i. $$\sqrt{-300}$$

Step-by-Step Solution

Answer

The expression \(\sqrt{-300}\) is equal to \(10i\sqrt{3}\).

Step 1: Identify the sqrt(-1) as i

Since the square root of a negative number is not defined in the real number system, we use the imaginary unit i, where i has the property that \(i^2=-1\). Hence, we can identify \(\sqrt{-1}\) in our problem with i.

Step 2: Separate the number -300

\(\sqrt{-300}\) can be separated into \(\sqrt{-1}\) and \(\sqrt{300}\) to apply the property that \(\sqrt{-1}\) is equal to i.

Step 3: Simplify the equation

After separating, we simplify the equation: \(\sqrt{-1} * \sqrt{300}\) = i * \(\sqrt{300}\). Next, \(\sqrt{300}\) can be simplified as it is a perfect square. \(\sqrt{300}\) can be simplified to \(10\sqrt{3}\). So, the expression becomes \(10i\sqrt{3}\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit is fundamental when dealing with the square roots of negative numbers. The imaginary unit is denoted as i, and it's defined by the property that i² = -1. This might seem unusual at first because in the real number system, every number squared gives a positive result. The imaginary unit allows for the extension of the real number system to the complex number system, which includes all possible linear combinations of a real number and an imaginary number.
Without the imaginary unit, we wouldn't be able to process the square roots of negative numbers, as these do not have solutions within the realm of real numbers. The introduction of i makes it possible to perform arithmetic operations that would otherwise be undefined, such as taking the square root of -300, and express them in terms of i.

Simplifying Square Roots

When it comes to simplifying square roots, the goal is to break down the expression into its simplest form to make it more understandable and easier to work with. This typically involves finding the prime factorization of the number under the square root and then using the property that the square root of a product equals the product of the square roots of the individual factors. For example, to simplify \(\sqrt{300}\), we look for perfect square factors of 300. Since 100 is a perfect square factor of 300 (100 times 3), and the square root of 100 is 10, we can rewrite \(\sqrt{300}\) as \(10\sqrt{3}\). This simplification clarifies the structure of the square root and reduces the complexity of the expressions we work with.

Radicals

The term radicals is another way to refer to square roots and other types of roots, like cube roots. They are represented by the symbol \(\sqrt{}\) for square roots, and with an index for higher roots, such as \(\sqrt[3]{}\) for cube roots. The key to working with radicals is understanding their properties, such as the aforementioned product property and the quotient property, which states that the square root of a quotient equals the quotient of the square roots. These rules are essential for performing operations with radicals and for simplifying complex expressions.

Negative Square Roots

The concept of negative square roots often confuses students because within real numbers, a negative number cannot have a square root. That's because squaring a real number always results in a non-negative product. However, once we extend to complex numbers, negative square roots become meaningful. A negative square root is expressed in terms of the imaginary unit i. For example, \(\sqrt{-300}\) is impossible to process if we only consider real numbers. But by identifying the square root of -1 as i, we can rewrite the expression as \(\sqrt{-300}\) = \(\sqrt{-1}\) * \(\sqrt{300}\), which simplifies to 10i\(\sqrt{3}\). The process involves associating each negative square root with an i to transition into the complex plane.