Problem 5 - Intermediate Algebra

Question

Fill in the blanks. The number 100 has two square roots. The positive or _____ square root of 100 is 10.

Step-by-Step Solution

Answer

The positive or **principal** square root of 100 is 10.

Step 1: Identify the Concept

A square root of a number is a value that, when multiplied by itself, gives the original number. There can be two square roots: one positive and one negative.

Step 2: Apply the Positive Square Root

The positive square root of a number is the non-negative root. For the number 100, the positive square root is often termed the principal square root.

Step 3: Understanding the Terminology

The term that commonly describes the non-negative square root is 'principal'. Therefore, the blank in the sentence should be filled with the word 'principal'.

Key Concepts

Positive Square Root

When you hear about the square root of a number, it's helpful to remember the positive square root first. The positive square root of a number is the non-negative value that, when multiplied by itself, gives the original number.
For instance, with the number 100, when you multiply 10 by itself, you indeed get 100 back. Thus, the positive square root of 100 is 10. This concept holds true for any non-negative number.
Understanding the positive square roots is beneficial, especially when dealing with real-life applications, such as determining the side length of a square with a given area in positive dimensions. Here, you'd use only the non-negative solution, often the one people first think of.

Principal Square Root

The term 'principal square root' is used interchangeably with 'positive square root'. It's specifically the non-negative root of a number. When we talk about the principal square root, we are focusing on the main or most common root taken by convention.
For example, for 100, as mentioned earlier, the principal square root is 10 because it's the non-negative square root that comes to mind immediately.

• The principal square root is most useful in situations where only positive values make sense, such as in physical measurements.
• It's also the standard root used in mathematical computations unless specified otherwise.

Understanding this term is vital for correctly using mathematical functions where non-negative results are required.

Negative Square Root

Now, let's explore the concept of negative square roots. While often less discussed, they are equally important. A negative square root is simply the negative counterpart of the positive square root.
Given any positive number, such as 100, though 10 is the principal square root, -10 is its negative square root. Why? Because -10 times -10 still equals 100.

• Negative square roots express numbers in a different range, which can be crucial for certain mathematical and scientific calculations.
• They're often used when solving equations where both positive and negative solutions are relevant.

Acknowledging that every non-zero positive number has two square roots—one positive and one negative—broadens our understanding of mathematical concepts beyond just the principal value.