Problem 38 - Intermediate Algebra

Question

Write each of the following in terms of \(i\) and simplify. $$ 5 \sqrt{-72} $$

Step-by-Step Solution

Answer

The expression simplifies to \(30i\sqrt{2}\).

Step 1: Understand the Problem

We need to express the number \(5 \sqrt{-72}\) in terms of \(i\). Remember that \(i\) is the imaginary unit where \(i^2 = -1\).

Step 2: Separate the Imaginary Unit

Since we have a negative number under the square root, we will separate it as \(\sqrt{-1} \times \sqrt{72}\). The \(\sqrt{-1}\) part is represented by \(i\).

Step 3: Break Down the Square Root

Next, we simplify \(\sqrt{72}\). The number 72 can be broken down into \(36 \times 2\), so \(\sqrt{72} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\).

Step 4: Combine the Parts

Combining everything together, the expression becomes \(5 \times i \times 6 \sqrt{2}\). We multiply these parts together to get \(30i\sqrt{2}\).

Step 5: Write the Final Answer

The simplified form of \(5\sqrt{-72}\) in terms of \(i\) is \(30i\sqrt{2}\).

Key Concepts

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part. The imaginary part involves the imaginary unit, represented by the symbol \(i\), where \(i^2 = -1\). This is the key characteristic that defines the imaginary unit. A complex number is generally expressed in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

• Real Part: This is the non-imaginary part of the complex number. In the expression \(a + bi\), \(a\) represents the real number part.
• Imaginary Part: The term \(bi\) is the imaginary part, where \(b\) is a real number coefficient and \(i\) is the imaginary unit.

Complex numbers are essential in mathematics because they extend the number system by providing solutions to equations that do not have real solutions. They allow for operations such as addition, subtraction, multiplication, and division just like real numbers, but they follow specific rules regarding the imaginary unit \(i\).
It's important to understand that any real number can be thought of as a complex number with no imaginary part, essentially writing \(a + 0i\). This concept helps in relating complex numbers to the rest of the number system.

Square Roots

A square root is a number that, when multiplied by itself, gives the original number. The square root of a positive number is fairly straightforward. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). Things become interesting when dealing with the square roots of negative numbers.When encountering a square root of a negative number, such as \(\sqrt{-1}\), we need to use the imaginary unit \(i\). By definition, \(\sqrt{-1} = i\). Therefore, any negative number under a square root can be re-expressed as the square root of a positive number times \(i\).To handle a square root of a negative, you can:

• First, factor the negative out, identifying \(\sqrt{-1}\), which becomes \(i\).
• Then, take the square root of the positive part, just like any standard square root.

For instance, with \(\sqrt{-72}\), we can express this as \(\sqrt{72} \cdot \sqrt{-1} = \sqrt{72} \cdot i\). Breaking down \(\sqrt{72}\) involves finding the square root of positive 72, which is \(\sqrt{36 \times 2} = 6\sqrt{2}\), and this combines to form \(6i\sqrt{2}\). Understanding these steps helps simplify expressions involving square roots of negative numbers into terms of \(i\).

Simplification of Expressions

Simplification of expressions is the process of reducing complex mathematical expressions into their simplest form. This often involves re-writing expressions to a form that is easier to understand or work with. When it comes to complex numbers and square roots, this process can include factoring, combining like terms, and applying known mathematical identities.
As in the given exercise, the expression \(5\sqrt{-72}\) needed simplification. Here is how simplification works in practice:

• Identify Parts: Recognize components, such as a square root of a negative number, that need to be rewritten using \(i\).
• Break Down Components: Use simpler mathematical operations, like factoring out perfect squares \(\sqrt{36}\) from \(\sqrt{72}\) to simplify \(\sqrt{72} = 6\sqrt{2}\).
• Apply Imaginary Unit: Remembering that \(\sqrt{-1} = i\) transforms the expression that originally included \(\sqrt{-72}\), turning it into \(5 \cdot i \cdot 6\sqrt{2}\).
• Combine: Multiply the numbers and terms together, as shown in the final multiplication resulting in \(30i\sqrt{2}\).

The goal is to create an expression that is manageable and follows mathematical conventions, making equations easier to work with in further calculations or in understanding the concepts represented by those expressions.