91Ó°ÊÓ

The power rule for exponents is a fundamental concept in algebra that makes simplifying expressions with exponents straightforward. It states that when you have an exponent raised to another exponent, you multiply them together. So, \((a^m)^n = a^{mn}\).
In our example, \((-3 x^{-6} y^{4})^{-2}\), we apply the power rule to each element inside the parentheses:
\((-3)^{-2} \, (x^{-6})^{-2} \, (y^{4})^{-2}\).
This step is crucial because it breaks down a complex expression into simpler parts, making the overall simplification process much easier. Always remember to multiply the exponents when applying the power rule. It is one of the most powerful tools you'll use when dealing with algebraic expressions.

Negative exponents can often seem confusing, but they become simple once you understand a key principle: a negative exponent indicates reciprocal. That means \(a^{-n} = \frac{1}{a^n}\).
This allows us to turn negative exponents into positive ones by taking the reciprocal of the base raised to the positive of that exponent. In our example, \((x^{-6})^{-2}\) is simplified by multiplying the exponents: \(-6 \times -2 = 12\). So, \((x^{-6})^{-2} = x^{12}\).
Similarly, \((y^{4})^{-2}\) becomes \(y^{-8}\). This transformation from negative to positive exponents is pivotal for further simplification.

Algebraic simplification is about reducing the expression to its simplest form. After addressing individual elements using the power rule and negative exponent rules, we focus on combining the results.
In our step-by-step solution:
1. We simplified each term as: \((-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}\), \((x^{-6})^{-2} = x^{12}\), and \((y^{4})^{-2} = y^{-8}\).
2. Next, we combine them: \(\frac{1}{9} \times x^{12} \times y^{-8}\).
3. Finally, after combining, we get: \(\frac{x^{12}}{9y^{8}}\).
This systematic approach ensures that no step is missed and helps convert a complicated-looking expression into an easily understood form.