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Factoring expressions is a critical skill in simplifying square roots and solving algebraic problems. It involves breaking down a complex expression into a product of simpler factors. This process makes it easier to identify and extract squares from under the square root, which can simplify the expression significantly.

For example, with the expression \(\sqrt{60 x^{5}}\), we start by factoring 60 into \(2^2 * 3 * 5\), which includes \(2^2\), a perfect square. The variable part, \(x^{5}\), is broken down into \(x^2 * x^2 * x\), where \(x^2\) are also perfect squares. Factoring expressions transforms the square root so that every perfect square factor can be easily simplified.

Understanding square root properties is vital for simplifying expressions containing square roots. One fundamental property to remember is that the square root of a perfect square \(a^2\) is simply \(a\). This property allows us to take pairs of factors that are perfect squares out from under the square root sign.

When we apply this property to the factored expression \(2^2 * 3 * 5 * x^2 * x^2 * x\), we can pull out \(2\) from \(2^2\) and \(x\) from each \(x^2\), leaving behind non-perfect squares under the square root. Proper application of square root properties simplifies the calculation and leads to a more manageable expression.

Algebraic simplification is the process of making an expression as simple as possible. This involves combining like terms and reducing the expression to its most concise form. After extracting perfect squares from under the square root sign, we focus on combining like terms outside the square root to achieve the simplest form.

In our example, multiplying the extracted terms \(2\) and \(x\) and \(x\) again gives us \(2x^2\). Then we rewrite the expression by bringing together the simplified parts and what remains under the square root as \(2x^2\sqrt{15x}\), which is the fully simplified form of the original expression. Mastery of algebraic simplification is key to efficiently tackling a wide variety of algebra problems.