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Prime factorization is a method of breaking down a composite number into a product of its prime factors. Primes are numbers that are greater than 1 and can only be divided evenly by 1 and themselves. For example, the prime factors of 48 are 2 and 3, which are both prime numbers. By repeatedly dividing the number by its smallest prime divisor, we find that 48 can be expressed as \( 2^4 * 3 \).
To simplify a square root using prime factorization, you look for perfect square factors within the prime factorization. For \(\sqrt{48}\), this involves recognizing that \( 2^4 \) is a perfect square (\( (2^2)^2 = 4^2 \) ), so \(\sqrt{48}\) can be simplified by taking the square root of the perfect square factor, resulting in \( 4\sqrt{3} \). This makes complex square root problems much more manageable.

Perfect squares are integers that are the product of an integer multiplied by itself. Recognizing perfect squares is crucial when simplifying square roots because the square root of a perfect square is always an integer. The square root function essentially asks, 'What number multiplied by itself gives me this value?' For instance, \( 81 \) is a perfect square since \( 9 * 9 = 81 \).
When dealing with square roots, identifying a perfect square allows for straightforward simplification, as seen with \(\sqrt{81}\), which simplifies directly to 9. Familiarizing yourself with common perfect squares (like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc.) can greatly speed up the process of simplifying radical expressions.

Radical expressions involve roots, such as square roots, indicated by the radical symbol (\( \sqrt{} \)). The goal when simplifying a radical expression is to find an equivalent expression where the radical is as simplified as possible. This often entails finding and extracting perfect squares from under the radical.
When simplifying the square root of a fraction, like \(\frac{\sqrt{48}}{\sqrt{81}}\), it's effective to simplify the square roots in the numerator and the denominator separately first, then simplify the fraction as a whole. Following this method, we can transform \(\sqrt{48}\) into \(4\sqrt{3}\) and recognize that \(\sqrt{81}\) is a perfect square, equal to 9. Put together, we then carry out the division to get \(\frac{4\sqrt{3}}{9}\). The beauty of working with radical expressions is that by using prime factorization and knowledge of perfect squares, even the most intimidating radical can be broken down into a more understandable form.