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. $$\sqrt{25}$$

Short answer

Answer: 5

Step-by-step solution

Identify the radicand

The radicand is the number inside the square root symbol, and in this case, it is 25.

Find the principal square root

We need to find a positive number that, when multiplied by itself, gives us 25. In this case, it is 5 because 5 * 5 = 25.

Write the simplified square root

Now that we know the principal square root of 25 is 5, we can write the simplified square root: $$\sqrt{25} = 5$$

Key concepts

Radicand

In mathematics, the term 'radicand' refers to the number or expression inside a radical symbol. In the context of square roots, specifically, the radicand is the value from which we wish to determine the square root.

For example, when we see the expression \(\root{2}{n}\), the 'n' is the radicand. It is the subject of our interest when simplifying a square root. Simplifying the square root means finding the number that, when multiplied by itself, equals the radicand. Therefore, understanding the radicand is the first and most crucial step in this process.

To improve comprehension, consider the radicand as the 'mystery' number you need to uncover by thinking of the square root as a question: 'Which number, when squared, gives me the radicand?' This approach helps conceptualize the relationship between square roots and their radicands.

Principal Square Root

The principal square root is defined as the non-negative square root of a number. When we write \(\root{2}{n}\), we're referring to the positive square root, also known as the principal square root. The number 'n' under the square root symbol, as discussed earlier, is the radicand.

The principal square root is particularly important because, for every positive number, there are actually two square roots: one positive and one negative. However, when we look at square roots in their simplest form, we default to the principal, or positive, solution because it is the principal value for real numbers.

Understanding that every positive number has two square roots can deepen our comprehension of the square root's nature. For example, although \(\root{2}{25}\) simplifies to 5, we acknowledge that \-5\ is also a square root of 25, but it is the negative square root, not the principal one.

Perfect Squares

Perfect squares are integral to simplifying square roots because they are the numbers that make this process straightforward. A perfect square is a number that can be expressed as the square of an integer. Some common perfect squares include 1 (\(1 \times 1\)), 4 (\(2 \times 2\)), 9 (\(3 \times 3\)), and so on.

In our exercise, the radicand 25 is a perfect square (\(5 \times 5\)), which simplifies the square root calculation. Recognizing perfect squares helps to quickly identify the principal square root without needing to perform any complex calculations.

Here is a list of the first few perfect squares that are helpful to remember:

• \(1^2 = 1\)
• \(2^2 = 4\)
• \(3^2 = 9\)
• \(4^2 = 16\)
• \(5^2 = 25\)

Mastering perfect squares can significantly aid in understanding and simplifying square roots, and they are the key to recognizing when a square root has been sufficiently simplified.