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The Power of a Power Rule is a fundamental principle when working with exponents. It helps in simplifying expressions that involve raising a power to another power. The rule is straightforward: If you have a term like \(a^{m}\) raised to the power of \(n\), it simplifies to \(a^{m \cdot n}\).

This means you multiply the exponents together to get the new exponent. Here’s a step-by-step approach to applying this rule:

• Identify the base and the exponents inside and outside the parentheses.
• Multiply the exponents: the exponent inside the parentheses by the one outside.
• Re-write the expression with the new exponent.

An example would be \(x^{3}\) raised to \(4\), which becomes \(x^{3 \cdot 4} = x^{12}\). By understanding this rule, you can simplify expressions more efficiently and avoid errors. In our problem, when applying this rule to \(x^{2}\) and \(y^{5}\), they become \(x^{2 \cdot 2} = x^{4}\) and \(y^{5 \cdot 2} = y^{10}\) respectively.

The Even Exponent Rule is very useful when dealing with negative numbers in exponential expressions. Understanding this rule helps you determine the sign of the resulting expression. The rule states: When a negative number is raised to an even exponent, the result is positive. This is because multiplying any negative number with itself an even number of times results in a positive product.

For instance, consider the expression \((-3)^2\). Here, \(-3\) is raised to the exponent \(2\), an even number. So, it becomes \(9\), because \(-3 \ imes -3 = 9\). Always remember this rule when you encounter a negative number raised to an even power, as it will help you simplify without getting stuck.

In the original problem, using this rule, \((-3)^2\) simplifies directly to \(9\), eliminating the negative sign and giving us a clean numerical coefficient to work with in further steps.

Simplifying expressions is often about breaking the problem down into smaller, manageable steps. Here’s a straightforward process that can be applied to exponential expressions:

• Begin by looking for any numerical parts that can be simplified using basic arithmetic.
• Apply any relevant exponent rules, such as the Power of a Power or Even Exponent rules.
• Organize your results, combining numerical coefficients and simplified variable expressions.

In our exercise, we initially handled the numerical part \((-3)^2\), converting it to \(9\) using the Even Exponent Rule. Then, we applied the Power of a Power Rule to simplify \(x^{2}\) and \(y^{5}\), ensuring the entire expression is simpler and easier to read: \(9x^{4}y^{10}\). This process of consistent, systematic simplification helps in solving complex expressions and builds confidence with exponential functions.