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Understanding how to simplify radical expressions is essential when dealing with square roots and higher roots in algebra. A radical expression is simplified when it meets certain conditions: the radicand (the number under the root sign) has no perfect square factors other than 1; the radicand is not a fraction; and no radicals appear in the denominator of a fraction.

To simplify a radical expression, begin by factoring out perfect squares from the radicand, and then apply the identity \( a^{1/2} \times a^{1/2} = a \) to any factors that are square roots. If the radical is part of a fraction, like our example \( \frac{3}{\sqrt{21}} \) from the original exercise, we would need to rationalize the denominator, which involves removing the radical from the denominator by multiplying both the numerator and the denominator by an appropriate form of 1 that includes the radical.

The concept of square roots is central to many algebraic processes, including the simplification of radical expressions. A square root, represented as \( \sqrt{x} \), answers the question, 'What number, when multiplied by itself, equals \( x \)?' The square root of a number is essentially half the exponent of squaring that number.

When dealing with square roots, one of the key skills is understanding the properties of radicals, such as \( \sqrt{x} \times \sqrt{x} = x \) and \( \sqrt{x \times y} = \sqrt{x} \times \sqrt{y} \) among others. In our initial exercise, rationalizing the denominator requires an understanding that multiplying \( \sqrt{21} \) by itself will give us the non-radical number, 21, which simplifies the expression.

Working with algebraic fractions, just like with numerical fractions, involves expressing quantities in terms of ratios. However, algebraic fractions can involve variables, exponents, and radicals. The principles of simplifying these fractions are alike; you seek to reduce the fraction to its simplest form by eliminating common factors from the numerator and the denominator.

In the case of our exercise, \( \frac{3}{\sqrt{21}} \) is an algebraic fraction with a radical in the denominator. The process we follow to rationalize the denominator and thus simplify the algebraic fraction is to create an equivalent fraction that does not have a radical in the denominator, without changing its value. The end goal is always to have a simplified expression that is easier to interpret and use in further calculations or applications.