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When dealing with expressions that include exponents applied to multiple bases within a parenthesis, such as \((a^{-2} y^{3})^{-3}\), we employ the *powers of products* rule. This rule states that if you have a product raised to an exponent, \((ab)^n\), it can be rewritten as \(a^n b^n\). This allows each part of the base to have the exponent applied individually.

In our example, \((a^{-2}y^{3})^{-3}\) is simplified by applying -3 to both \(a^{-2}\) and \(y^{3}\). Therefore, it becomes \(a^{-2*-3}y^{3*-3} = a^{6}y^{-9}\).

Using this rule helps break down complex expressions into simpler parts that are easier to manage, especially when combined with other rules of exponents.

The *quotient rule of exponents* comes handy when simplifying the division of like bases with exponents. Consider the expression \(\frac{a^m}{a^n}\), you can simplify it by applying the rule \(a^{m-n}\). This saves time and reduces the complexity of expressions.

In our problem, \(\frac{a^{6}y^{-9}}{a^{2}y}\), we apply this rule separately to each base:

• For \(a\), apply the rule: \(a^{6-2} = a^{4}\).
• For \(y\), apply the rule: \(y^{-9-1} = y^{-10}\).

The result \(a^{4}y^{-10}\) is a more streamlined form of the fraction. This technique effectively reduces the exponents, simplifying the expression further.

Negative exponents are often misunderstood but are a pivotal concept in simplifying expressions with exponents. A negative exponent suggests the reciprocal of the base raised to the positive of that same exponent. Simply put, \(a^{-n}\) can be rewritten as \(\frac{1}{a^n}\). This conversion is crucial for removing negative exponents from the expression.

Take \(a^{4}y^{-10}\), for instance. While \(a^4\) is already in its simplest form, \(y^{-10}\) needs to be rewritten as its reciprocal. Thus, it becomes \(\frac{1}{y^{10}}\), leading to a final simplified version of the expression: \(a^4 \times \frac{1}{y^{10}}\) or, more concisely, \(\frac{a^4}{y^{10}}\).

Understanding negative exponents is essential for grasping overall exponent rules and keeping expressions clear and properly simplified.