Problem 46 - Intermediate Algebra

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \sqrt{-7} \sqrt{-10} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\sqrt{70}\).

Step 1: Express Each Root in Terms of i

Both \(\sqrt{-7}\) and \(\sqrt{-10}\) involve the square root of a negative number. We know that \( \sqrt{-1} = i \). Therefore, \( \sqrt{-7} = \sqrt{7} \cdot i \) and \( \sqrt{-10} = \sqrt{10} \cdot i \).

Step 2: Multiply the Expressions

Multiply the expressions \( \sqrt{7} \cdot i \) and \( \sqrt{10} \cdot i \) together. This gives us: \( (\sqrt{7} \cdot i) \cdot (\sqrt{10} \cdot i) = \sqrt{7} \sqrt{10} \cdot i^2 \).

Step 3: Simplify the Expression

Simplify \(\sqrt{7} \sqrt{10} \cdot i^2\). We know that \(i^2 = -1\), so the expression becomes \(-\sqrt{7} \cdot \sqrt{10}\).

Step 4: Final Simplification

Calculate \(\sqrt{7} \cdot \sqrt{10}\) which results in \(\sqrt{70}\). Thus, the expression simplifies to \(-\sqrt{70}\).

Key Concepts

Imaginary Unit

When we talk about the imaginary unit, we're referring to a very special number in mathematics denoted as \(i\). It represents the square root of \(-1\).
This concept is crucial because, in the real number system, taking the square root of a negative number is not possible.
The imaginary unit \(i\) allows us to extend numbers beyond the real line into what we call the complex plane. Here are some key points about the imaginary unit:

• The fundamental property of \(i\) is that \(i^2 = -1\).
• It enables us to express square roots of negative numbers in terms of real numbers and \(i\). For example, \(\sqrt{-4}\) is written as \(2i\) since \(\sqrt{4} = 2\) and \(\sqrt{-1} = i\).
• Complex numbers are written in the form \(a + bi\), where \(a\) and \(b\) are real numbers. In this sense, \(i\) acts like a coordinate showing the position on the imaginary axis.

Understanding \(i\) is crucial to grasping the broader concept of complex numbers, where both real and imaginary parts come together.

Square Roots of Negative Numbers

Finding square roots of negative numbers might seem unusual because normal arithmetic with real numbers doesn't support it. However, the imaginary unit \(i\) allows us to handle this. Here's how it works:When you face the square root of a negative number \(\sqrt{-x}\) (where \(x\) is positive), you would express it using the imaginary unit:\[\sqrt{-x} = \sqrt{x} \cdot i\]
This expression breaks down the problem into the familiar square root of a positive number and the square root of \(-1\), symbolized by \(i\).What does this achieve? It transforms a non-real number into something we can work with mathematically:

• The square root of a negative number becomes manageable and fits within the structure of complex numbers.
• This transformation helps perform operations by using the property \(i^2 = -1\) to simplify further calculations.
• By breaking down \(\sqrt{-7}\) and \(\sqrt{-10}\) into \(\sqrt{7} \cdot i\) and \(\sqrt{10} \cdot i\), we can proceed with multiplication and simplification.

Understanding how to correctly write and simplify the square roots of negative numbers using \(i\) is essential when dealing with complex numbers and their operations.

Complex Multiplication

Complex multiplication involves multiplying numbers that include the imaginary unit \(i\). This technique is similar in concept to multiplying polynomials, where you multiply coefficients and follow distribution across terms.To multiply two complex numbers or expressions like \(\sqrt{7} \cdot i\) and \(\sqrt{10} \cdot i\) from our problem, you follow these steps:

• Multiply the numerical (real) parts together. Here, it's \(\sqrt{7} \cdot \sqrt{10}\), which results in \(\sqrt{70}\).
• Multiply the imaginary units \(i \cdot i\) to apply the property \(i^2 = -1\).
• The combination \((\sqrt{7} \cdot i) \cdot (\sqrt{10} \cdot i)\) simplifies to \(\sqrt{70} \cdot i^2\), which further reduces to \(-\sqrt{70}\).

Each of these steps relies heavily on the fundamental properties of \(i\), such as how \(i^2\) yields \(-1\), turning the expression involving the square root of negative numbers into a simpler form.By mastering complex multiplication, you unlock the ability to manipulate and solve a wide array of problems in mathematics and engineering that involve complex numbers.