Problem 42 - Intermediate Algebra

Question

Write each of the following in terms of \(i\) and simplify. $$ 9 \sqrt{-40} $$

Step-by-Step Solution

Answer

The simplified expression is \(18\sqrt{10}i\).

Step 1: Identify the Expression Type

The expression given is \(9 \sqrt{-40}\), which involves a square root of a negative number. Recall that the imaginary unit \(i\) is defined as \(\sqrt{-1}\). This will help us rewrite the square root of the negative number in terms of \(i\).

Step 2: Rewrite the Square Root

Start by rewriting \(\sqrt{-40}\) as \(\sqrt{40} \times \sqrt{-1}\). This uses the property that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) when both \(a\) and \(b\) are non-negative. This gives us \(\sqrt{40} \times i\).

Step 3: Simplify the Radical

Next, simplify \(\sqrt{40}\). Note that \(40 = 4 \times 10\), where \(4\) is a perfect square. So, \(\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}\).

Step 4: Substitute Back and Simplify

Now substitute \(\sqrt{40} = 2\sqrt{10}\) back into the expression from Step 2. We have \(\sqrt{-40} = 2\sqrt{10} \times i\). Multiply this result by 9 as given in the problem: \(9 \times 2\sqrt{10} \times i = 18\sqrt{10}i\).

Step 5: State the Final Simplified Expression

The fully simplified expression in terms of \(i\) is \(18\sqrt{10}i\). Therefore, \(9\sqrt{-40} = 18\sqrt{10}i\).

Key Concepts

Imaginary Unit

Numbers can sometimes be difficult to work with, especially when they involve taking the square root of a negative number. In this context, the imaginary unit "i" is a helpful tool. By definition, the imaginary unit is represented as\(i = \sqrt{-1}\). This allows us to work with numbers that we couldn't otherwise. For example, the square root of \(-4\) can be expressed as \(2i\), because \(-4 = 4 \times (-1)\). Anytime you have the square root of a negative number, you can express it using "i". This concept is fundamental in simplifying complex numbers, allowing you to transform otherwise challenging calculations into more manageable ones.By using the imaginary unit, you open up a world of mathematical possibilities. Complex numbers, which include real and imaginary parts, are extensively used in fields such as engineering, physics, and computer science.

Simplifying Radicals

Simplifying radicals, particularly when working with complex numbers, involves reducing expressions to their simplest form. Radicals refer to expressions that contain a square root, cube root, or other root symbols. To simplify, identify factors within the radicand, which is the number inside the root symbol.For example, if you have \(\sqrt{40}\), break it down into the product of its factors: \(\sqrt{4 \times 10}\). Since \(4\) is a perfect square, it can be simplified. Thus, \(\sqrt{4} \times \sqrt{10} = 2\sqrt{10}\). This process of breaking down the radicand into manageable parts and simplifying them is crucial. Some helpful tips include:

• Look for perfect square factors in the radicand, as these can be simplified easily.
• Rewrite the radicand as a product of simpler numbers if possible.

Simplifying radicals is not only about making numbers easier to manage but also about transforming complex number expressions into their simplest forms for easier calculations.

Multiplication of Imaginary Numbers

Multiplying imaginary numbers involves combining the values of the terms and simplifying the expression accordingly. When you multiply two imaginary numbers or expressions involving "i", apply normal arithmetic rules but remember that \(i^2 = -1\).Let's consider the multiplication of \(9\sqrt{-40} \). By rewriting the term, we have \(9 \times 2\sqrt{10} \times i \), resulting from the simplified form as we previously mentioned. Calculate it step-by-step:

• Multiply the coefficients: \(9 \times 2 = 18\).
• Combine the terms with the radical: \(18\sqrt{10}\).
• Do not forget the \(i\): \(18\sqrt{10}i\).

The process shows how the multiplication of imaginary numbers integrates the principles of typical arithmetic operations, but adjusted for the presence of the imaginary unit. This understanding is crucial when calculating complex expressions and transforming them into simpler forms, allowing their practical application in various scientific and engineering contexts.