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Prime factorization involves breaking down a number into its basic building blocks, known as prime numbers. Consider the number inside the radical. With, for instance, the number 200, we need to express it as a product of prime numbers.
We do this by dividing it by as many times as possible by the smallest prime number until all factors left are prime.

• Start with the smallest prime: 200 divided by 2 gives us 100.
• 100 divided by 2 gives us 50.
• 50 divided by 2 gives us 25.
• Finally, 25 can be divided by 5 since 5 is a prime number: 25 divided by 5 gives us 5, and 5 divided by 5 results in 1.

The prime factorization of 200 thus is expressed as: \[ 200 = 2^3 \cdot 5^2 \]This representation is fundamental, especially when simplifying radicals.

Simplifying radicals requires understanding certain properties they possess. A radical can be simplified using the following basic properties:
When you have a radical expression like \( \sqrt{a^{2}} = a \), it tells us that when we take a square root of a square number, we get the base number.
More generally, for even indexed radicals: \( \sqrt[n]{a^{n}} = a \).
Here's how these properties come into play:

• If you have \( \sqrt{5^2} \), using the property, this simplifies directly to 5.
• In cases like \( \sqrt{y^2} \), likewise, this results in y.

These properties help "release" perfect squares from under the radical, aiding in simplifying the expression further.

The radicand is the expression inside a radical sign. In our problem, the radicand is \( 200xy^2 \). Let's explore how different components of the radicand affect the process of simplification:

• Number part: The number 200 is represented in its prime factors as \( 2^3 \cdot 5^2 \).
• Variable part: \( x \) and \( y^2 \) remain part of the radicand, affecting the simplification based on their powers.

When simplifying, even powers of variables, like \( y^2 \), allow extraction outside the radical since they can be entirely simplified. Odd powers or simple variables without an exponent, like \( x \), typically remain within the radical. Hence, the radicand's nature directly influences what emerges out of simplification.

In a radical expression, the coefficient is the number placed outside the radical sign that multiplies the simplified form. Originally, our given expression was \[ 5 \sqrt{200xy^{2}} \].
The 5 outside is the initial coefficient. During the simplification process, other numbers get pulled out, becoming part of a new coefficient.

• Firstly, during simplification, from \( \sqrt{200} \), a 5 and a 2 get pulled out (from \( 5^2 \) and \( 2^2 \)).
• The \( y \) also emerges out of simplification as a factor alongside these numbers.

So the initial 5 multiplies with these factors, creating a new coefficient. In the end, we combine everything to ensure that our expression remains in its simplest radical form. Here, the coefficient becomes 250, as \( 5 \times 5 \times 2 = 50 \) with \( y \), leading to \( 250y \sqrt{2x} \). This demonstrates how vital the coefficient is in presenting the fully simplified radical expression.