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The power rule is a fundamental idea in algebra that helps us to simplify expressions involving exponents. It states that when you raise a power to another power, you can multiply the exponents together. This can be expressed as \( (a^m)^n = a^{m \times n} \). This rule is very useful when dealing with expressions inside parentheses that are raised to a power. For instance, in the exercise \((4 x^{2} y)^{3}\), you apply the power rule to break down each component separately. The constant 4, the \(x^2\), and the \(y\) all need to be raised to the power of 3. Thus, the expression turns into \(4^{3} \times (x^2)^3 \times y^3\). Using the power rule simplifies complex expressions, making it easier to manage and ultimately solve them.

Simplifying expressions involves breaking down complicated mathematical phrases into simpler, more manageable parts. This often requires using rules and properties like the power rule, distributing multiplications, and combining like terms. In our example, after applying the power rule to \( (4 x^{2} y)^{3} = 4^{3} \times (x^2)^3 \times y^3 \), the next step is to simplify each part separately:

• For the constant: Calculate \( 4^3 = 64 \).
• For the variable \( x \): Apply the power rule to get \( (x^2)^3 = x^{6} \).
• For the variable \( y \): It's already simplified as \( y^3 \).

Combining these simplified parts, we get \( 64x^6y^3\). Simplifying expressions helps to solve equations more easily and to understand the relationships between different variables and constants within them.

Prealgebra serves as the foundation for all higher levels of mathematics. It involves basic arithmetic, understanding integers, fractions, decimals, and the properties of numbers. Concepts such as the power rule and simplifying expressions are all rooted in prealgebra. For instance, knowing how to multiply and raise powers comes straight from prealgebra lessons. Prealgebra teaches students the principles they need to solve more complex algebraic equations. By mastering these core skills, one can better handle more advanced topics like exponentiation, polynomial equations, and eventually calculus. Remember, each little step in prealgebra builds your math muscles for higher-level math!