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A square root is a value that, when multiplied by itself, gives the original number. The concept of square roots is fundamental in mathematics because it helps simplify numerical expressions. In this exercise, we encounter the square root of a number, specifically \( \sqrt{20} \).Square roots can be challenging to simplify directly, so we break them down into easier parts. For example, with 20, we factor it into 4 and 5 since 4 is a perfect square. This helps because \( \sqrt{20} \) can be written as \( \sqrt{4} \times \sqrt{5} \). Knowing that \( \sqrt{4} \) simplifies to 2, we transform \( \sqrt{20} \) into \( 2\sqrt{5} \).In solving mathematical problems, identifying and simplifying such square roots is essential for arriving at a neat and manageable expression.

Variable expressions involve letters representing numbers. These variables can often be more complex to work with when inside square roots or other functions. In this problem, we focus on \( \sqrt{x^3} \).To simplify variable expressions under a square root, apply the rule \( \sqrt{x^n} = x^{(n/2)} \). This step involves calculating the half of the exponent, which, for \( x^3 \), becomes \( x^{3/2} \). Further simplification turns \( x^{3/2} \) into \( x \cdot \sqrt{x} \).When dealing with variable expressions in equations, it is important to transform them into simpler forms, making calculations easier and results clearer.

Exponent rules guide us in handling numbers raised to powers, such as variables. Simplifying expressions like \( x^3 \) under a root involves using these rules efficiently.A fundamental rule is when you take the square root of a power, you essentially divide the exponent by 2. For example, in \( \sqrt{x^3} \), the exponent rule transforms it to \( x^{3/2} \). Then, it can split into \( x \cdot \sqrt{x} \), separating whole numbers from root expressions.Combining exponent rules with square roots is critical in simplifying complex algebraic expressions, as seen in this exercise. Mastering these rules results in more accurate and elegant solutions in mathematics.