Problem 47 - Intermediate Algebra

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \sqrt{-9} \sqrt{-6} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-3\sqrt{6}\).

Step 1: Express each square root in terms of 'i'

The expression involves the square roots of negative numbers, which can be expressed in terms of the imaginary unit 'i'. Recall that the square root of a negative number \(-a\) is \( ext{i} imes ext{sqrt}(a)\).Thus,\(\sqrt{-9} = \sqrt{9} \times i = 3i\) and \(\sqrt{-6} = \sqrt{6} \times i = \sqrt{6}i\).

Step 2: Multiply the expressions

Now that both expressions are in terms of 'i', we perform the multiplication:\(\sqrt{-9} \times \sqrt{-6} = (3i) \times (\sqrt{6}i)\).

Step 3: Simplify the expression

Multiply the coefficients and simplify:- Multiply the real numbers: \(3 \times \sqrt{6} = 3\sqrt{6}\)- Multiply the imaginary parts: \(i \times i = i^2 = -1\)Therefore, the expression simplifies to:\((3\sqrt{6})(-1) = -3\sqrt{6}\).

Key Concepts

Square Roots

Square roots are operations that find the number which, when multiplied by itself, yields the original number. However, finding the square root of negative numbers requires special treatment. This is where the concept of imaginary numbers comes in. An imaginary number is a number that can be written as a real number multiplied by the imaginary unit 'i', where the key characteristic of 'i' is that it satisfies the equation \(i^2 = -1\).

When dealing with the square roots of negative numbers, as in our exercise, the expression can be rewritten in terms of 'i'. For example, the square root of a negative number \(-a\) can be expressed as \(\sqrt{-a} = i \cdot \sqrt{a}\). This allows us to handle operations involving square roots of negative numbers more easily.

Understanding this principle is crucial for solving complex number problems correctly, as it transforms seemingly unsolvable expressions into manageable forms.

Simplifying Expressions

Simplifying expressions, especially when dealing with imaginary numbers, involves breaking down and reducing the expression to its simplest form. The prime goal is to eliminate as much complexity as possible without changing the expression’s overall value.

In our example, after rewriting square roots using 'i', we need to simplify further by multiplying these expressions. Simplification can follow a few key steps:

• Express each part of the expression clearly in terms of real numbers and 'i'.
• Perform arithmetic operations such as multiplication or distribution normally but keep track of the imaginary unit rules (\(i^2 = -1\)).
• Combine like terms wherever feasible to reach a simpler form.

This process is essential for arriving at a final answer in its simplest possible form. For our problem, once the multiplication is completed, recognizing and applying \(i^2 = -1\) helps simplify the result to \(-3\sqrt{6}\).

Remember, the better you simplify, the easier it is to work with the expression later on.

Multiplication of Complex Numbers

When multiplying complex numbers, especially those involving the imaginary unit 'i', it is important to apply the distributive property and rules for operations correctly. In our problem, we have two parts \( (3i) \) and \( (\sqrt{6}i) \) that need to be multiplied.

This multiplication involves:

• Multiplying real coefficients together, which gives us \(3 \times \sqrt{6} = 3\sqrt{6}\),
• Multiplying the imaginary units (i \times i) together, which leads to \(i^2\) that equals \(-1\).

Adding these multiples together is crucial to solve the expression completely. As seen, we applied that \(i^2 = -1\) rule to convert the expression \((3\sqrt{6})(-1)\) simplifying it to \(-3\sqrt{6}\).

Understanding the multiplication of complex numbers is vital, as it helps streamline the process and ensure accuracy in arithmetic with imaginary numbers.