Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. Simplify: $$ \sqrt{-36}=\sqrt{\cdot 36}=\sqrt{36}=6 $$

Step-by-Step Solution

Answer

Fill in the blanks with: -1 and 6i.

Step 1: Identify the Expression

We begin with the expression \( \sqrt{-36} \). This represents the square root of a negative number, which involves the imaginary unit \( i \).

Step 2: Factor Out \( -1 \)

Recognize that \( -36 \) can be rewritten as \( (-1) \times 36 \). This allows us to separate the real part from the imaginary component.

Step 3: Apply the Property of Square Roots

Use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Here, \( \sqrt{-36} = \sqrt{-1 \times 36} \) becomes \( \sqrt{-1} \times \sqrt{36} \).

Step 4: Simplify Each Part

The square root of \( -1 \) is \( i \), and the square root of 36 is 6. Thus, we have: \( \sqrt{-1} \times 6 = i \times 6 = 6i \).

Key Concepts

Imaginary Numbers

In mathematics, imaginary numbers are used to assist us in finding square roots of negative numbers, which is otherwise impossible within the realm of regular real numbers. The concept of imaginary numbers arises naturally in situations like the one we are discussing, where we need to evaluate a square root of a negative number.

• The imaginary unit is represented by the symbol \( i \).
• By definition, \( i = \sqrt{-1} \).

When dealing with negative numbers under the square root, we use the imaginary unit to "factor out" the negative aspect. For example, \( \sqrt{-36} \) can be rewritten as \( \sqrt{(-1) \times 36} \). Here, the \(-1\) is addressed by the \( i \), letting us express \( \sqrt{-1} = i \). Thus, \( \sqrt{-36} \) becomes \( i \times \sqrt{36} \).
By leveraging imaginary numbers, we can extend our understanding and solve problems that would otherwise remain unsolvable when stuck to only real numbers.

Square Roots

The square root of a number is an essential mathematical function that essentially reverses the operation of squaring a number. It's denoted by the radical symbol \( \sqrt{} \). When we discuss the square root of a positive number, it refers to a value that, when multiplied by itself, gives the original number back. For example, \( \sqrt{36} = 6 \) because \( 6 \times 6 = 36 \).
However, taking the square root of a negative number introduces complexity because no real number multiplied by itself produces a negative number. To address this, imaginary numbers come into play. When you see an expression like \( \sqrt{-36} \), you're compelled to factor as \( \sqrt{-1 \times 36} \) and apply the property: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Through this property and the definition of \( i \), you can solve square roots of negative numbers, as in \( \sqrt{-36} = i \times 6 = 6i \).
The art of decomposing numbers and understanding these properties is foundational to mastering complex numbers and mathematical operations.

Simplification of Expressions

Simplification of expressions involves breaking down a more complicated expression into an easier-to-read form without changing its value or meaning. This process is crucial in managing expressions in both algebraic and complex numbers realms.
Let's consider the expression \( \sqrt{-36} \). The simplification starts with recognizing what makes the expression complex or challenging. Here, the negative number under a square root immediately signals the use of imaginary numbers. By rewriting \( -36 \) as \((-1) \times 36\), we can separately evaluate the real part (\(36\)) and the imaginary part (\(-1\)). This separation helps further break down the expression.

• Once broken down, apply fundamental properties like \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
• The square root of \(-1\) is converted to \( i \), and for positive numbers like \( 36 \), it translates to its simple square root, here \( 6 \).

The final simplified form, \( 6i \), presents the original complex expression in a tidier format. Mastery of such simplifications allows deeper mathematical insights, clears complex expressions quickly, and is essential for advanced theories and applications.