Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$\sqrt{-5} \cdot \sqrt{-10}$$

Step-by-Step Solution

Answer

-5 sqrt(2)

Step 1: Express numbers as complex numbers

Express \(\sqrt{-5}\) and \(\sqrt{-10}\) as complex numbers. Hence we can write \(\sqrt{-5} = \sqrt{5} \cdot \sqrt{-1} = i \sqrt{5}\) and \(\sqrt{-10} = \sqrt{10} \cdot \sqrt{-1} = i \sqrt{10}\) respectively. So, our expression becomes: \(i \sqrt{5} \cdot i \sqrt{10}\)

Step 2: Multiply the complex numbers

Recalling that when multiplied, 'i' behaves just like any other variable, and \(i^2 = -1\), we get: \(i \sqrt{5} \cdot i \sqrt{10}\) = \(i^2 \sqrt{50}\) = \(- \sqrt{50}\).

Step 3: Simplify the Result

The square root of \(50\) can be simplified as \(\sqrt{25 \cdot 2}\) = \(5 \sqrt{2}\). Hence, our expression becomes \(-5 \sqrt{2}\).

Key Concepts

Multiplying Complex Numbers

Multiplying complex numbers can initially seem daunting, but it becomes straightforward once you understand the process. Essentially, the multiplication involves expanding and simplifying expressions where both real and imaginary components are present. When multiplying two complex numbers, each element with a real and imaginary part is distributed, and like terms are combined.

Consider two complex numbers in their standard form: \(a + bi\) and \(c + di\). Their multiplication follows from the formula \( (ac - bd) + (ad + bc)i \), where \(a, b, c,\) and \(d\) are real numbers, and \(i\) is the imaginary unit.

This method of expansion and collection of like terms is exemplified further in multiplying numbers such as \(i \sqrt{5} \cdot i \sqrt{10}\) from the exercise. Here, recognize that \(i \cdot i = i^2\), which simplifies the expression since \(i^2 = -1\). Understanding this behavior is crucial in simplifying expressions.

Imaginary Unit

The imaginary unit, represented as \(i\), is the cornerstone of complex numbers. It defines numbers that are not real since \(i\) is the square root of minus one. Therefore, conventionally, we know that \(i^2 = -1\). This characteristic becomes particularly significant in operations such as multiplication.

In our exercise, expressing \(\sqrt{-5}\) and \(\sqrt{-10}\) using the imaginary unit aids us in tackling square roots of negative numbers. By setting \(\sqrt{-1} = i\), it becomes possible to write any imaginary number in the form \(bi\), where \(b\) is a real number.

Understanding \(i\)'s properties illuminates how complex numbers operate, and especially aids in operations like multiplication and division, which are fundamental for students looking to master complex mathematical concepts.

Simplifying Radicals

Simplifying radicals is an essential skill in algebra that allows us to express roots in their simplest form. When you encounter a radical, consider if it can be broken down further into standard numbers. This process, called simplification, mainly involves expressing the number under a root in terms of its prime factors.

Take for instance \(\sqrt{50}\), found in our solution. Breaking this down, recognize that \(50 = 25 \times 2\), allowing you to simplify this radical as \(5 \sqrt{2}\). By acknowledging \(25\)'s square root \(5\), we extract a coefficient from the radical, simplifying the expression.

This method aids in not only making equations easier to manage but also in providing a form that's more aesthetically pleasing and correct for further mathematical operations. Therefore, mastering this skill is highly valuable in efficiently solving higher-level math problems.