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Understanding how to simplify radical expressions is essential for solving various algebra problems. A radical expression involves roots, like square roots or cube roots. Simplifying them often makes it easier to perform further algebraic operations.

To simplify a radical expression, you should look for perfect squares—or cube roots when dealing with cube radicals—and extract them outside the radical. For example, \(\sqrt{16} = 4\) because 16 is a perfect square. If you have expressions like \(\sqrt{3+2\sqrt{2}}\) and \(\sqrt{3-2\sqrt{2}}\) under radicals, look for a way to simplify or combine them in a manner that removes the radical. This process might include 'rationalizing the denominator', which we will cover in the next section.

In our exercise, the expression inside the radical did not contain perfect squares, but by clever manipulation using the rationalization technique, we were able to simplify the expression into a non-radical form: 2.

Rationalizing the denominator is a technique used to eliminate radicals from the denominator of a fraction. The purpose of this process is to create a simpler, rational expression. When a radical is present in the denominator, the expression is often considered not fully simplified.

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial \(a + b\) is \(a - b\), and vice versa. This works because, when a binomial is multiplied by its conjugate, the result is the difference of squares: \(a + b)(a - b) = a^2 - b^2\), which eliminates the radical.

Example in Practice

In our original exercise, multiplying by the conjugate transformed the expression with radical denominators into a simple integer. We essentially used the identity \(\sqrt{a}+\sqrt{b}\) times \(\sqrt{a}-\sqrt{b}\) to turn it into \(a - b\), which is free of radicals and much easier to manage. Thus, the expression \(\sqrt{3+2\sqrt{2}} - \sqrt{3-2\sqrt{2}}\) was simplified to 2.

In mathematics, a perfect square is an integer that is the square of another integer. For instance, the number 9 is a perfect square because it equals \(3^2\). Recognizing perfect squares is useful because their square roots are rational, and they simplify to whole numbers.

When dealing with radical expressions, identifying whether the number under the radical is a perfect square can immediately tell you if the expression simplifies to a rational or an irrational number. So, knowing your perfect squares can save a lot of time in problem-solving.

How Perfect Squares Relate to Our Problem

In the exercise's solution, after simplifying the expression for \(x\), we squared the result to find \(x^2\). This calculation yielded 4, which is a perfect square (\(2^2\)). Therefore, the solution to \(x^2\) is a rational number. This is a key concept because it implies that any root of a perfect square results in a rational number, and thus the original expression \(x\) is rational as well.