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Prime factorization is the process of breaking down a composite number into a product of its prime numbers. A prime number is a natural number greater than 1, which cannot be formed by multiplying two smaller natural numbers. Every number has a unique prime factorization.
For instance, the number 45 can be broken into its prime factors by dividing it by the smallest prime number 3, which it is divisible by (since it ends in a 5). When 45 is divided by 3, it results in 15, which can be further divided by 3, resulting in 5. 5 is a prime number. Therefore, 45 can be expressed as:

• 45 = 3 × 3 × 5
• 45 = 3² × 5

This factorization is crucial because it makes simplifying square roots much easier when using the Square Root Property.

The Square Root Property aids in simplifying expressions involving square roots. It states that, for any non-negative integers,\[\sqrt{a^2 imes b} = a \cdot \sqrt{b}.\]This property is quite handy when dealing with prime factorizations. Imagine we've got a radicand that includes a perfect square (like 3² in our case).The Square Root Property allows you to take the square root of the perfect square effortlessly.
In our example from the solution, we have: \ \(\sqrt{3^2 \times 5} = 3 \sqrt{5}\).
We notice that \(3^2\) under the square root becomes just 3 when moved outside the radical. This drastically simplifies the expression.

Simplifying radicals is about making square roots easier to work with by reducing them to their simplest form. To simplify a radical, like \(\sqrt{45}\), you first make use of prime factorization, then the Square Root Property to take factors out of the radical symbol.
For instance:

• Begin by expressing 45 as its prime factors: \(3^2 \times 5\).
• Apply the Square Root Property: \(\sqrt{3^2 \times 5}=3\sqrt{5}\).
• Replace \(\sqrt{45}\) in the original expression with \(3\sqrt{5}\).
• Multiply the coefficients outside the radical to simplify further.

So, going from \(\frac{3}{4} \sqrt{45}\) to \(\frac{9}{4} \sqrt{5}\) makes computations more manageable while retaining the same value!