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When squaring radical calculations, as seen in the exercise where we demonstrate that ewline \(\sqrt{9-4 \sqrt{5}}=\sqrt{5}-2\), ewline the process requires the individual to deal carefully with exponents and roots. This specific action, squared by raising to the power of 2, effectively removes the square root and reveals what is underneath the radical. In the initial step, both sides of the original equation are squared. ewline This leads to: ewline \[(\sqrt{9-4 \sqrt{5}})^2 = (\sqrt{5}-2)^2\] ewline It's crucial to ensure that while squaring a binomial—a two-term algebraic expression—like ewline \(\sqrt{5}-2\), ewline we utilize the FOIL (First, Outside, Inside, Last) method or the distributive property to correctly expand and square the terms. The squaring of radical equations often includes this step of expanding expressions to reveal a more straightforward form to work with.

A key element in the study of algebra is the ability to simplify square roots, which admits stripping the root down to its simplest form without changing its value. Simplifying can also involve the elimination of any square root in the denominator of a fraction or reducing a square root by splitting into two factors, one of which is a perfect square. ewline In the provided solution, squaring both sides made simplification possible, resulting in the simplified equation ewline \[9 - 4\sqrt{5} = 9 - 4\sqrt{5}\] ewline This result confirms the equality. Simplifying square roots is not just a mechanical process; it involves the recognition of patterns and an understanding of number properties, like the fact that the square root of a product equals the product of the square roots.

When we talk about 'surds' in algebra, we refer to roots that are left in their radical form, usually because they cannot be simplified into integer or fractional numbers. Surds are irrational numbers, and they retain their exact values because they are not expressed as decimals. ewline Getting comfortable with surds means understanding that operations involving them can often lead to a further simplification or a rationalization where the surd is moved from the denominator to the numerator. ewline The exercise ewline \(\sqrt{9-4 \sqrt{5}}\) ewline may initially look like a surd that can't be easily simplified. Yet, through the process of squaring, we find it equals to the expression ewline \(\sqrt{5}-2\), ewline illustrating a crucial characteristic of surds: they hold surprises and can sometimes be simplified more than they initially appear to be.