Problem 43 - Intermediate Algebra

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \sqrt{-4} \sqrt{-16} $$

Step-by-Step Solution

Answer

-8

Step 1: Express square roots in terms of i

The expression \( \sqrt{-4} \) can be rewritten as \( \sqrt{4} \cdot \sqrt{-1} \). Since \( \sqrt{4} = 2 \) and \( \sqrt{-1} = i \), we get \( \sqrt{-4} = 2i \). Similarly, \( \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i \).

Step 2: Multiply the expressions

Now that we have \( \sqrt{-4} = 2i \) and \( \sqrt{-16} = 4i \), multiply these results: \[ (2i)(4i) = 8i^2 \]

Step 3: Simplify using properties of i

Recall that \( i^2 = -1 \). Substitute \( i^2 \) in the expression:\[ 8i^2 = 8(-1) = -8 \]

Key Concepts

Imaginary Unit

Complex numbers come with a magical twist, known as the imaginary unit, which is represented by the symbol \( i \). The imaginary unit is the building block for working with square roots of negative numbers, and it fundamentally helps in the expansion of the real number system.

• The defining property of \( i \) is that \( i^2 = -1 \).
• This means that \( i \) is essentially the square root of \( -1 \), or \( \sqrt{-1} = i \).

With \( i \), we can take the square root of negative numbers, which was previously an impossible task when only dealing with real numbers. Through this concept, complex numbers are formed, which combine both real and imaginary parts.

Square Roots of Negative Numbers

Encountering a negative number under a square root might seem challenging, but the imaginary unit \( i \) simplifies this process. When dealing with expressions like \( \sqrt{-4} \) or \( \sqrt{-16} \), we rewrite these as products of positive numbers and \( i \):

• \( \sqrt{-4} \) becomes \( \sqrt{4} \cdot \sqrt{-1} = 2i \).
• \( \sqrt{-16} \) transforms to \( \sqrt{16} \cdot \sqrt{-1} = 4i \).

By rewriting the square roots of negative numbers in terms of \( i \), they become much easier to handle. This way, calculations can proceed using the properties of imaginary numbers, without any conceptual blocks.

Simplifying Expressions

After expressing numbers with \( i \), simplification becomes the key to understanding complex results, especially when dealing with multiplication.
In the problem context:

• First, we rewrite \( \sqrt{-4} \) and \( \sqrt{-16} \) in terms of \( i \) as \( 2i \) and \( 4i \) respectively.
• Then, multiply these results: \( (2i)(4i) = 8i^2 \).

The final step involves utilizing the property \( i^2 = -1 \). Substitute this into the expression to get:\

• \( 8i^2 = 8(-1) = -8 \).

This process shows how complex expressions can be reduced to simple numeric terms by using the properties of the imaginary unit effectively.