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Prime factorization is the process of expressing a number as a product of its prime numbers. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In the context of simplifying square roots, prime factorization is an essential step.

Consider the number 27. To factor it, look for the smallest prime number, which is 2, and see if it's a factor. Since 27 is not even, move to the next prime number, which is 3. The number 27 can be divided by 3, and the quotient is 9. Do the same to 9, which can be divided by 3 again, with a quotient of 3. Continue this process until only the number 1 remains.

As a result, 27 can be expressed as a product of its prime factors as follows:

• 27 = 3 × 3 × 3 = 3³

This means 27 is composed entirely of the prime number 3. This factorization will help in the next steps to simplify the square root.

The square root property is a crucial algebraic tool that allows us to simplify square roots more easily. The property states that \( \sqrt{a^2b} = a\sqrt{b} \). This property proves highly useful when dealing with numbers that can be broken down into squares.

In the case of \( \sqrt{27} \), after the prime factorization, we have \( 3^3 \). To apply the square root property, we break it into \( 3^2 \times 3 \). According to the property, this can be expressed as \( \sqrt{3^2 \times 3} \), allowing us to take the square root of \( 3^2 \).

This simplifies to \( 3\sqrt{3} \), as the square root of \( 3^2 \) is 3, which makes the \( \sqrt{3} \) remain under the radical. This means the square root of a perfect square multiplies outside the square root, reducing complexity in calculations.

Simplifying square roots involves combining prime factorization and the square root property to make the expression neater and more understandable. Let’s reiterate the steps we took to simplify \( \sqrt{27} \).

• First, find the prime factors of the number inside the square root. For 27, this is \( 3 \times 3 \times 3 = 3^3 \).
• Next, use the square root property. Rewrite the expression as \( 3^2 \times 3 \) under the square root, \( \sqrt{3^2 \times 3} \).
• Then, apply the property: take the square root of \( 3^2 \) outside, resulting in \( 3\sqrt{3} \).

Thus, these steps transform the original form into a simplified radical form. This process illustrates how recognizing patterns in numbers and understanding properties can simplify complex expressions effectively.