Problem 34 - A Graphical Approach to College Algebra

Question

Multiply or divide as indicated. Simplify each answer. $$\sqrt{-17} \cdot \sqrt{-17}$$

Step-by-Step Solution

Answer

The simplified answer is -17.

Step 1: Recognize the multiplication of square roots

This exercise involves multiplying two square roots of negative numbers: \( \sqrt{-17} \cdot \sqrt{-17} \).

Step 2: Apply the property of square roots

Use the property that states \( \sqrt{a} \cdot \sqrt{a} = a \). Thus, \( \sqrt{-17} \cdot \sqrt{-17} = -17 \).

Step 3: Simplify the expression

The expression simplifies directly to \( -17 \) since \( \sqrt{-17} \cdot \sqrt{-17} = -17 \) using the property.

Key Concepts

Square Roots

Square roots are numbers that, when multiplied by themselves, give the original number under the square root. For instance, the square root of 9 is 3, since 3 times 3 equals 9.
When dealing with square roots of negative numbers, it's important to remember that these involve imaginary numbers, represented by the symbol \( i \) where \( i = \sqrt{-1} \).

• \( \sqrt{-17} \) is actually \( \sqrt{17} \times i \), since \( \sqrt{-1} = i \).
• Thus, regarding square roots of negative numbers, they involve performing operations with imaginary components.

Understanding this concept sets the foundation for dealing with more complex scenarios such as products of square roots.

Multiplication of Square Roots

Multiplying square roots can often seem daunting, but it follows a simple property: \( \sqrt{a} \cdot \sqrt{a} = a \).
This states that the square root of a number, squared, equals the number itself.

• In our example, \( \sqrt{-17} \times \sqrt{-17} \) becomes \( (-1) \times 17 = -17 \).
• This utilizes the property that \( \sqrt{-1} \times \sqrt{-1} = -1 \), reinforcing the use of imaginary numbers \( i \).

It is crucial to be aware of these underlying rules to simplify and effectively tackle such operations. This approach helps in solving similar kinds of problems in algebra and assures you are applying properties correctly.

Simplification

Simplification is a key step in algebra that involves reducing expressions to their simplest form.
It allows us to express results in more straightforward and clear terms.

• Starting with \( \sqrt{-17} \times \sqrt{-17} \), we apply the property described above to directly simplify it to \( -17 \).
• This simplification affirmatively concludes your solution and presents the result in its simplest numeric form.
• The essence of simplification is to make these seemingly complex expressions more manageable and understandable.

By mastering the steps of simplification, students can handle various algebraic problems with ease and clarity, ensuring they not only arrive at the correct answer but fully understand the process that led them there. This is a vital skill in all branches of math, acting as the bridge between complex problems and their solutions.