Problem 24 - Elementary Algebra

Question

Perform the operations. $$ 6 i(1-2 i) $$

Step-by-Step Solution

Answer

The result is \( 12 + 6i. \)

Step 1: Distribute the Scalar

First, distribute the scalar value of the coefficient outside the parentheses, which is 6i, to each term inside the parentheses. This means you multiply 6i with 1 and 6i with -2i separately.

Step 2: Multiply the Terms

Calculate \(6i \cdot 1 = 6i\) and \(6i \cdot (-2i) = -12i^2.\)

Step 3: Simplify the Complex Number

Recall that \(i^2 = -1\). Substitute \(-1\) for \(i^2\) in the expression \(-12i^2\) to get \(-12(-1) = 12.\)

Step 4: Combine Terms

Combine the real number and the imaginary number obtained: \(12 + 6i.\) This is the result of the operation.

Key Concepts

Imaginary Unit

Complex numbers are made up of two parts: a real part and an imaginary part. The imaginary part involves the imaginary unit, denoted as \(i\). The defining property of \(i\) is that \(i^2 = -1\). This unique property helps us handle operations involving square roots of negative numbers, which aren't possible with real numbers alone. For example, \(\sqrt{-1} = i\).

When multiplying complex numbers or simplifying expressions, using this property is key. For instance, if you encounter \(-12i^2\) during an operation, remember that \(i^2\) equals \(-1\). Thus, \(-12i^2\) becomes \(-12(-1)\), leading to \(12\).

Understanding the imaginary unit \(i\) is essential for delving deeper into complex numbers and simplifying their expressions.

Distributive Property

The distributive property is a core tool in algebra that allows us to simplify expressions, especially when involving parentheses and multiplication. It states that if you have an expression in the form of \(a(b + c)\), you can distribute \(a\) across the terms inside the parentheses, resulting in \(ab + ac\). This property also applies neatly to complex numbers.

For example, in the expression \(6i(1-2i)\), the distributive property lets you multiply \(6i\) by each term inside the parentheses. So, \(6i \cdot 1 = 6i\) and \(6i \cdot -2i = -12i^2\).

Use this property to make complex multiplications more manageable. This way, you break down expressions into simpler parts, making them easier to solve or simplify.

Simplification of Expressions

Simplifying expressions involves techniques that reduce expressions to their simplest form. For complex numbers, simplification often includes using the properties of the imaginary unit and distributing terms as seen in our exercise.

Start by applying the distributive property to break down complex multiplication into simpler parts. For instance, with \(6i(1-2i)\), the distribution yields terms \(6i\) and \(-12i^2\).

Next, replace \(i^2\) with \(-1\). Doing so turns \(-12i^2\) into \(12\), which simplifies the expression significantly.

• Combine the real and imaginary parts resulting from your simplification.
• Look for like terms and perform necessary operations.

So, from \(12 + 6i\), you effectively get the simplest form of the expression initially given. This process ensures clarity and incredibly simplifies the handling of complex numbers.