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When you encounter an expression like \((3x^2y^3)^2\), the power of a product property becomes very useful. This mathematical property states that when a product of several terms is raised to an exponent, you apply that exponent to each term in the product individually.

This means you treat the numbers and variables inside the parenthesis separately. In our expression, the number 3, and the variables \(x^2\) and \(y^3\) all get raised to the power of 2. This gives us \(3^2\), \((x^2)^2\), and \((y^3)^2\).

By doing so, you make the solving process systematic and manageable. Remember, it's like sharing the outside exponent with each part of the inside terms.

Exponents are a way to represent repeated multiplication. When dealing with positive exponents, you're essentially multiplying the base by itself several times. Positive exponents make calculations clear and consistent. They are straightforward because they only involve multiplication.

In our problem, we transformed each term into its simplest form using positive exponents:

• \(3^2 = 9\) simply means multiplying the number 3 by itself once.
• For the variable \(x\), we have \((x^2)^2 = x^{2 \times 2} = x^4\). This indicates \(x\times x \times x \times x\), or four x's multiplied together.
• Lastly, \((y^3)^2 = y^{3 \times 2} = y^6\), meaning six y's in a product.

Remember to always express answers with positive exponents for clarity and convention.

Simplifying an expression like \((3x^2y^3)^2\) involves breaking down the expression into its simplest components. This makes it easier to work with in equations or further calculations.

Through following the simplification process, first, apply the power of a product property to ensure each component is addressed individually with the exponent. Then, calculate each power. This involves converting numbers and variables into their simplest form using positive exponents:

• \(3^2 = 9\), a straightforward multiplication.
• \(x^4\) from \((x^2)^2\) simply shows keeping count of the repeated multiply.
• \(y^6\) from \((y^3)^2\) further reduces the complexity.

Finally, combining these results gives a fully simplified expression: \(9x^4y^6\). Simplification is helpful as it removes complexity and focuses on essential elements for understanding and solving mathematical problems.