Question

Prep Exercise \(11 \quad \sqrt{n}\) indicates the ___ square root (principal square root) of the radicand, \(n .\) Prep Exercise \(12-\sqrt{n}\) indicates the ____ square root of the radicand, \(n\) simplify. $$\sqrt{64}$$

Short answer

Answer: 8

Step-by-step solution

Understand the problem

We are given a square root expression, and we need to find the principal square root of the number 64.

Recall the property of square roots

Remember that the square root of a number is a value that, when multiplied by itself, gives the original number. In other words, if \(a = \sqrt{b}\), then \(a \times a = b\) or \(a^2=b\).

Find the principal square root

Now, we will find the principal square root of 64 by determining which number, when squared, equals 64. We can quickly see that 8, when squared, gives us 64. Therefore, the principal square root of 64 is 8.

Write the final answer

So, the simplified expression for the given problem is: $$\sqrt{64}=8$$

Key concepts

Principal Square Root

When we talk about square roots, the term "principal square root" refers to the non-negative square root of a non-negative number. In mathematics, the square root symbol, \(\sqrt{}\), is used to denote the principal square root. We rely on principal square roots to simplify expressions and make calculations.

It's important to note that every positive number actually has two square roots: one positive and one negative. However, the principal square root prioritizes the positive value. So, when you see \(\sqrt{64}\), it inherently means the principal square root, which is 8.

Whenever you're working with square roots, especially in algebra or geometry problems, remember that unless explicitly stated, you're expected to find the principal square root, i.e., the positive one. Understanding this can help avoid confusion particularly in school homework or exams.

Radicand

In any square root expression, the radicand is simply the number or expression inside the square root symbol \(\sqrt{}\). The radicand is crucial because it determines what number's root you are calculating.

Let's break it down with our example: in \(\sqrt{64}\), the number 64 is the radicand. It is the number that we are seeking to find the square root of.

Here are a few key points about radicands:

• If the radicand is positive, the result of the principal square root is a real number.
• If the radicand is zero, the square root is zero.
• If the radicand is negative, it results in an imaginary number, since you cannot determine a real number that squares to give a negative result.

Thus, understanding the radicand helps you figure out what the square root operation will yield. Paying attention to the radicand can also alert you to potential errors if the number is not ideal for simplification.

Simplification of Square Roots

Simplifying square roots is all about finding the most reduced form of the square root expression without altering its value. The process of simplification may involve identifying any perfect squares within the radicand and using their square roots to streamline the expression.

For instance, simplifying \(\sqrt{64}\) involves recognizing that 64 is a perfect square because it can be rewritten as \(8^2\). As a result, the square root of 64 simplifies directly to 8.

Here are some steps you can follow when simplifying square roots:

• Identify any perfect square factors within the radicand.
• Express the radicand as the product of a perfect square and another factor.
• Take the square root of the perfect square factor, simplifying the expression.

This process is useful for making complex square root expressions easier to work with. Simplifying square roots is a fundamental skill in solving many algebraic equations and understanding mathematical relationships.