91Ó°ÊÓ

Radicals, represented by the square root symbol (√), are used to denote the square root of a number. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \(\text{sqrt}(16) = 4\), because \(4 \times 4 = 16\).

Understanding radicals helps in solving a variety of mathematical problems, including simplifying expressions and solving equations. When you see a fraction under a radical, you can simplify it by first simplifying the fraction and then taking the square root of the numerator and the denominator separately.

For example, in the problem \( \text{sqrt}\bigg(\frac{4}{9}\bigg)\), we notice both the numerator (4) and the denominator (9) are perfect squares. Therefore, we can simplify it to \(\frac{2}{3}\).

A perfect square is a number that is the square of an integer. In other words, it is a number that can be expressed as \( n^2 \) where \( n \) is an integer.

Examples of perfect squares include:

• 4 (since \(2^2 = 4\))
• 9 (since \(3^2 = 9\))
• 16 (since \(4^2 = 16\))

Recognizing perfect squares can make simplifying radicals easier. For instance, in the exercise, we simplify the fraction \( \frac{4}{9} \) by identifying 4 and 9 as perfect squares. This simplification leads to:
\( \text{sqrt}\bigg(\frac{4}{9}\bigg) = \frac{\text{sqrt}(4)}{\text{sqrt}(9)} = \frac{2}{3} \).

Fraction simplification is the process of making a fraction as simple as possible. When dealing with radicals involving fractions, this simplification becomes very handy.

For example, if we have the fraction \( \frac{4}{9} \), we recognize that both the numerator (4) and the denominator (9) are perfect squares. We can rewrite the fraction under the radical as:

\( \text{sqrt}\bigg(\frac{4}{9}\bigg) = \frac{\text{sqrt}(4)}{\text{sqrt}(9)} = \frac{2}{3} \).

This result shows how useful simplification is. Instead of dealing with a more complex radical, we break it down into simpler, manageable parts.

A negative square root is simply the negative value of the principal (positive) square root. For instance, the principal square root of 4 is 2. Therefore, the negative square root of 4 is -2.

In the context of the given exercise, we are dealing with \-\text{sqrt}(0.01)\. Recognizing that \(0.01 = 0.1^2\), we find that the positive square root of 0.01 is 0.1:

\(\text{sqrt}(0.01) = 0.1\).

To apply the negative sign, we simply write:

\(-\text{sqrt}(0.01) = -0.1\).

It's important to remember that when a square root is negative, it means the final answer should carry the negative sign, just like in our example.