Problem 13 - Algebra A Combined Function

Question

Multiply or divide as indicated. $$ \sqrt{16} \cdot \sqrt{-1} $$

Step-by-Step Solution

Answer

The result is \(4i\).

Step 1: Simplify the Square Root of 16

The square root of 16 can be simplified, as it is a perfect square. Therefore, \( \sqrt{16} = 4 \).

Step 2: Identify the Square Root of Negative One

The square root of \(-1\) is defined as the imaginary unit \(i\). That is, \(\sqrt{-1} = i\).

Step 3: Multiply Simplified Radicals

After simplifying, we multiply the results from Steps 1 and 2: \( 4 \times i = 4i \).

Key Concepts

Square Roots

Square roots are numbers that, when multiplied by themselves, produce a given number. For example, the square root of 16 is 4 because 4 times itself equals 16. Not every number has a simple square root like 16. When dealing with figures under a square root symbol, we are usually looking to find this value.

• Perfect Squares: Numbers like 1, 4, 9, 16, 25 are perfect squares since they are the result of squaring whole numbers.
• Calculating: To find the square root is like asking "What value multiplied by itself gives this number?"

Understanding square roots helps when working with equations and algebra because it is a foundational skill used frequently in simplifying expressions and solving problems.

Imaginary Unit

The imaginary unit, denoted as \(i\), is crucial in mathematics, especially when dealing with square roots of negative numbers. The imaginary unit \(i\) is defined by the equation \(i^2 = -1\). This definition allows us to expand our number system to include "imaginary numbers."

• Imaginary Numbers: Numbers that involve \(i\), such as \(i\), \(2i\), \(3 + 4i\).
• Purpose: Imaginary numbers are essential for solving equations like \(x^2 + 1 = 0\), where a solution doesn't exist within the real numbers alone.

Using the imaginary unit involves a mindset shift, as it means accepting that some numbers can have representations that don't exist in the traditional sense. Yet, they allow for a broader range of mathematical applications and solutions.

Multiplication of Radicals

Multiplication of radicals is a process where we multiply terms that are within square root symbols. When multiplying radicals like \(\sqrt{a} \cdot \sqrt{b}\), the product rule for roots is typically applied.

• Product Rule: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
• Practical Example: In the original exercise, we had \(\sqrt{16} \cdot \sqrt{-1}\), where we first simplified \(\sqrt{16}\) and recognized \(\sqrt{-1}\) as \(i\).

Applying multiplication rules allows us to condense and solve expressions more effectively. This skill is especially helpful when both radicals are simplified, resulting in a straightforward multiplication of their products.