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A radical is a mathematical symbol that represents the root of a number. The most common type is the square root, denoted by the symbol \( \sqrt{} \). When you see \( \sqrt{a} \), it means the number that, when multiplied by itself, gives you \(a\). For example, \( \sqrt{9} \) equals 3 because 3 multiplied by 3 is 9.

Radicals are used in various mathematical contexts, including solving equations and simplifying expressions. In the given exercise, \( \sqrt{3} \) is a radical.

Recognizing and understanding radicals is crucial for simplifying expressions and performing operations on them. When you encounter radicals, remember that you can often simplify them by finding a number that can be multiplied by itself to give the radicand (the number inside the radical). For instance, \( \sqrt{16} = 4 \), because 4 times 4 equals 16.

Simplifying fractions means reducing them to their simplest form. A fraction is simplified when the numerator and the denominator have no common factors other than 1. In the context of our exercise, we simplify the fraction \( \frac{\sqrt{3}}{2+\sqrt{3}} \) by rationalizing its denominator.

Rationalizing the denominator refers to the process of eliminating the radical from the denominator. This typically involves using the conjugate of the denominator. The conjugate of a binomial \(a + b\) is \(a - b\).

Here's how we apply it:

• Multiply both the numerator and the denominator by the conjugate of the denominator. In our case, the conjugate of \(2 + \sqrt{3} \) is \(2 - \sqrt{3} \).
• This gives us \( \frac{\sqrt{3}}{2+\sqrt{3}} * \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{\sqrt{3}(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} \).
• Next, we perform the multiplication in the numerator: \(\sqrt{3}(2 - \sqrt{3}) = 2\sqrt{3} - 3 \).
• For the denominator, we use the difference of squares: \(2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1 \).

This simplifies our fraction to \(\frac{2\sqrt{3} - 3}{1} = 2\sqrt{3} - 3\). The fraction is now in its simplest form.

The difference of squares is a useful algebraic identity used to simplify expressions. It states that for any two terms \(a\) and \(b\), \(a^2 - b^2 = (a + b)(a - b)\). This identity is particularly handy when working with radicals and rationalizing denominators.

In our exercise, we used the difference of squares to simplify the denominator \((2 + \sqrt{3})(2 - \sqrt{3})\). Here’s how:

We recognize that \((2 + \sqrt{3})(2 - \sqrt{3})\) fits the pattern \(a^2 - b^2\), where \(a = 2\) and \(b = \sqrt{3}\).

• Applying the difference of squares, we get \(2^2 - (\sqrt{3})^2\).
• This simplifies to \(4 - 3 = 1\).

Using this identity simplifies multiplication and helps eliminate radicals from the denominator. It is a powerful tool in algebra, helping to simplify complex expressions and making calculations more manageable.