Problem 41 - Intermediate Algebra

Question

Write each of the following in terms of \(i\) and simplify. $$ 12 \sqrt{-90} $$

Step-by-Step Solution

Answer

The simplified form is \(36i \sqrt{10}\).

Step 1: Express the Negative Square Root Using i

We start by expressing the negative square root in terms of the imaginary unit \(i\). Recall that \(\sqrt{-1} = i\). Thus, \(\sqrt{-90} = \sqrt{-1 \times 90} = \sqrt{-1} \times \sqrt{90} = i \times \sqrt{90}\).

Step 2: Simplify the Radical

Next, simplify \(\sqrt{90}\). Notice that 90 can be factored as \(9 \times 10\), where 9 is a perfect square. Hence, \(\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}\).

Step 3: Combine the Results

Substitute the simplified square root back into the expression:\(12 \times i \times 3\sqrt{10} = (12 \times 3)i\sqrt{10} = 36i\sqrt{10}\).

Key Concepts

Square Roots

Square roots are mathematical operations that help us find a number which, when multiplied by itself, gives us the original number. Typically, square roots are symbolized using the radical sign \(\sqrt{}\). For example, \(\sqrt{25} = 5\) because \(5 \times 5 = 25\).

When dealing with negative numbers under the square root sign, we encounter imaginary numbers. This is because no real number squared will yield a negative number. Here, the imaginary unit \(i\) is introduced, defined by \(i = \sqrt{-1}\).

This allows us to simplify negative square roots by rewriting them in terms of \(i\). For instance, if you need to solve \(\sqrt{-90}\), you can express this as \(\sqrt{-1 \times 90} = i \times \sqrt{90}\). This transformation is essential in working with complex numbers, combining real and imaginary components.

Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form. We achieve this by identifying and extracting perfect square factors from the radicand (the number under the square root).

For example, when simplifying \(\sqrt{90}\), recognize that 90 can be factored into \(9 \times 10\). Since 9 is a perfect square (\(\sqrt{9} = 3\)), we can extract it out of the square root. Therefore, \(\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}\).

Simplifying radicals helps make complex expressions more manageable. Extracting the square factor often turns an expression into a simpler or more intuitive form. Remember this process whenever you deal with more complicated radical expressions.

Complex Numbers

Complex numbers are numbers that incorporate both a real part and an imaginary part. They are typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the imaginary unit.

In the exercise \(12 \sqrt{-90}\), the negative square root is transformed into a complex number. We convert it using \(i\), resulting in \(12i\sqrt{90}\). This expression is then simplified and further rearranged to \(36i\sqrt{10}\), showcasing an expression with a real coefficient \(36\) and an imaginary component \(i\sqrt{10}\).

Understanding complex numbers is important because they extend the real number system. They allow us to perform mathematical operations even when negative square roots are present. Complex numbers represent a wide array of mathematical phenomena, including electrical engineering, physics, and advanced mathematics.