Problem 42 - Algebra for College Students

Question

Write each of the following in terms of \(i\) and simplify. For example, $$ \sqrt{-20}=i \sqrt{20}=i \sqrt{4} \sqrt{5}=2 i \sqrt{5} $$ $$9 \sqrt{-40}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Negative Under the Square Root

Recognize that the expression involves taking the square root of a negative number. Here, we are given \(9 \sqrt{-40}\), and we know that \(-40\) is negative.

Step 2: Express as a Product Involving \(i\)

Use the property that \(\sqrt{-1} = i\). Therefore, we can write \(\sqrt{-40} = \sqrt{-1 \times 40}\), which simplifies to \(i \sqrt{40}\). So now we have \(9 \times i \sqrt{40}\).

Step 3: Simplify the Square Root

Next, simplify \(\sqrt{40}\). We can factor 40 into 4 and 10 since 4 is a perfect square. Thus, \(\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10}\).

Step 4: Simplify the Expression

Substitute \(\sqrt{40}\) with \(2\sqrt{10}\) in the expression. We now have \(9 \times i \times 2 \sqrt{10} = 18i \sqrt{10}\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit is fundamental when working with complex numbers. In mathematics, the imaginary unit is denoted by the letter \(i\), and it is defined as the square root of \(-1\). This means that \(i^2 = -1\). This concept might seem strange at first, because no real number squared gives a negative result, but \(i\) provides a way to work with these previously "impossible" numbers.

Imaginary numbers allow us to expand our number system and solve equations otherwise unsolvable with real numbers alone, such as those involving square roots of negative values. For instance, when you see a square root involving a negative number, like \(\sqrt{-40}\), you can rewrite it in terms of \(i\).

• Real parts and imaginary parts: In complex numbers, a number can have a real component and an imaginary component, such as \(a + bi\).
• Complex plane: Imaginary numbers can be represented on a perpendicular axis to the real numbers, forming a complex plane.

Square Roots

Taking square roots is a common operation in algebra, yet becomes interesting when working with negative numbers. Normally, the square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). But with negative numbers, the real number system doesn't offer solutions. This is where complex numbers take a role.

For negative square roots, you utilize the imaginary unit \(i\). For example, consider \(\sqrt{-40}\). This can be rewritten as \(\sqrt{-1 \times 40}\), allowing us to separate the square root of \(-1\) as \(i\), making it possible to handle the positive component separately. So, we express \(\sqrt{-40}\) as \(i\sqrt{40}\).

Then, you can simplify further. Recognize the perfect square factor: 40 can be broken down into 4 and 10. Hence, \(\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}\). This allows for simpler expressions and easier calculations.

Simplification Steps

Simplification is a crucial aspect of understanding and working with complex numbers. This process involves breaking down expressions to their simplest form. With the expression \(9\sqrt{-40}\), we aim to simplify this into a form involving \(i\) and rational numbers.

Initially, recognize the square root of the negative part involves \(i\). Break it down as discussed: \(\sqrt{-40} = i\sqrt{40}\). Next, address the square root: \(40\) splits into \(4\) and \(10\), specifically choosing \(4\) because it is a perfect square, allowing \(\sqrt{40} = \sqrt{4} \cdot \sqrt{10} = 2 \sqrt{10}\).

• Substitution and multiplication: Substitute back into the multiplication \(9 \times i \times 2 \sqrt{10} = 18i \sqrt{10}\)
• Consistently check your work: Simplifying step-by-step reduces the possibility of error and clarifies complex expressions.

By following these steps, complex expressions become much more manageable, creating clarity and understanding.