Problem 41 - Algebra for College Students

Question

Write each of the following in terms of \(i\) and simplify. For example, $$ \sqrt{-20}=i \sqrt{20}=i \sqrt{4} \sqrt{5}=2 i \sqrt{5} $$ $$12 \sqrt{-90}$$

Step-by-Step Solution

Answer

\(12 \sqrt{-90} = 36i \sqrt{10}\)

Step 1: Recognize the Imaginary Unit

Identify the imaginary unit associated with the negative number under the square root. Here, we have \(-90\), which indicates the use of the imaginary unit \(i\) where \(i = \sqrt{-1}\).

Step 2: Factor the Negative Number

Write \(-90\) as \(-1 \times 90\). This allows us to separate the negative part from the positive for the use of the imaginary unit.

Step 3: Apply the Imaginary Unit

Since \(\sqrt{-90} = \sqrt{-1 \times 90}\), it becomes \(\sqrt{-1} \times \sqrt{90}\). Using \(\sqrt{-1} = i\), it becomes \(i \sqrt{90}\).

Step 4: Simplify the Square Root

Factor \(90\) into a product of perfect squares: \(90 = 9 \times 10\). Thus, \(\sqrt{90} = \sqrt{9} \times \sqrt{10} = 3 \times \sqrt{10}\).

Step 5: Combine the Results

Combine the imaginary unit with the simplified square root: \(12 \sqrt{-90} = 12 \times i \times 3 \times \sqrt{10} = 36i \sqrt{10}\).

Key Concepts

Complex Numbers

Complex numbers might sound a bit complicated at first because they involve an imaginary unit, but they are not that difficult once you break them down. A complex number is a number that has both a real part and an imaginary part. The standard form is written as:

• Real part: a
• Imaginary part: bi (where i is the imaginary unit, which equals \(\sqrt{-1}\)).

An example of a complex number is 3 + 4i. It’s like taking a regular number and adding a bit of creativity! The imaginary unit i is the foundation when dealing with complex numbers. It's important because it lets us take square roots of negative numbers, which wasn't possible with just real numbers.Here’s what you need to know about complex numbers:

• They extend our numeric system beyond the real numbers.
• They enable us to effectively work with and solve equations that don't have real solutions.

Square Roots

The square root is another essential concept when dealing with both real and complex numbers. Simply put, finding a square root is like reverse multiplication. When you take the square root of a number, you're finding a value which, when multiplied by itself, gives that number back.Consider the square root of a positive number, like 9. It’s straightforward: \(\sqrt{9} = 3\) because \(3 \times 3 = 9\). But what about negative numbers? This is where the imaginary unit i comes into play. We can't find the square root of a negative number without using i because there is no real number that, when squared, will result in a negative.That’s why \(\sqrt{-90}\) was rewritten as \(i\sqrt{90}\). Separation of components inside the square root sign allows us to manage with real numbers by setting aside the negative part for i.

Simplification

Simplification in mathematics, especially when dealing with square roots, helps make expressions easier to work with. The goal is to get to the simplest form without altering the value.When it comes to expressions involving square roots of negative numbers, like in our example \(12\sqrt{-90}\), simplification involves several steps:

• First, recognize that there's an imaginary component. \(\sqrt{-1}\) separates as i.
• Next, decompose the components of the square root into simpler parts.
• Identify any perfect squares, which are numbers like 4, 9, or 16 — numbers that have whole number square roots.
• The expression \(\sqrt{90}\) gets broken down into \(3\sqrt{10}\) because \90 = 9 \times 10\ and \(\sqrt{9} = 3\).
• Finally, gather the results and multiply them, ensuring you're following mathematical operations order.

So when you simplify, you are really tidying up the expression to make it presentable.

Algebraic Expressions

Algebraic expressions, like the one involved in \(12\sqrt{-90}\), are basic tools in algebra involving variables, numbers, and operations together. Think of them as mathematical phrases that can include constants (like the 12), variables (like x or y), coefficients (numbers in front of variables), and even imaginary units.By breaking down the algebraic expression you've got:

• The constant 12, which stays the same.
• The operation of multiplication connecting all parts — the root, the number, and the imaginary unit.
• Finally, the simplified square root, as previously calculated.

Working through an algebraic expression is like organizing LEGO blocks. You have to fit all pieces together correctly to build a complete structure, ensuring each piece is in its rightful place. This includes recognizing when a part can or cannot be simplified further, what roles each part plays, and how they operate together as a complete unit.