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Perfect square factorization is the process of breaking down a number into factors, where one of those factors is a perfect square. This is helpful when simplifying square roots because perfect squares have simple square roots. A perfect square is a number that can be expressed as the product of an integer with itself. For instance, 1, 4, 9, 16, and 25 are all perfect squares:

• 1 = 1 \( \times \) 1
• 4 = 2 \( \times \) 2
• 9 = 3 \( \times \) 3
• 16 = 4 \( \times \) 4
• 25 = 5 \( \times \) 5

In the exercise of simplifying \( \sqrt{80} \), we can identify that 16 is the largest perfect square that divides 80 without a remainder. Factor 80 as 16 \( \times \) 5. Now, we can take the square root of each factor separately, and this leads us to the simpler expression below:

• \( \sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} \)

Square roots are a fundamental concept in mathematics, particularly when it comes to simplifying expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, denoted as \( \sqrt{16} = 4 \), because 4 \( \times \) 4 equals 16. When dealing with square roots, if a number is not a perfect square, the radical expression may need to be simplified using factorization.In the problem \( \sqrt{80} \), the aim is to simplify the square root expression as much as possible. By applying perfect square factorization (identifying 16 as a factor), we express \( \sqrt{80} \) as \( \sqrt{16} \times \sqrt{5} \). This uses the principle that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Finally, \( \sqrt{16} \) simplifies to 4, resulting in \( 4 \times \sqrt{5} \). This is now in its simplest form, showcasing an important aspect of algebraic simplification.

Algebraic simplification is the process of reducing expressions to their simplest form while preserving their values. It involves combining like terms, reducing numbers, and applying mathematical operations in a logical sequence. Simplifying square root expressions is a common task in algebra. The exercise primarily focuses on taking a complex radical expression like \( \sqrt{80} \) and rewriting it in a simplified form using the tools mentioned above.For \( \sqrt{80} \), algebraic simplification starts by decomposing 80 using perfect square factors. After identifying that 80 can be factored into 16 and 5, we apply the rule \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \) to simplify further. After breaking it down, \( \sqrt{16} \) simplifies directly to 4 as 16 is a perfect square. Therefore, \( \sqrt{80} \) simplifies to \( 4\sqrt{5} \), which cannot be reduced further. This final expression is the simplest representation of the original square root.