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The power rule is a central concept in simplifying algebraic expressions, particularly when dealing with exponents. The rule states that \( (a^m)^p = a^{mp} \). This means when you raise a power to another power, you multiply the exponents. In the context of the exercise, we applied the power rule to individual terms inside the parentheses. For example, \( (q^4)^3 \) becomes \( q^{4 \times 3} = q^{12} \). This simplifies complex expressions and makes it easier to handle larger equations.
By understanding and using the power rule correctly, you can simplify parts of the algebraic expression efficiently and prepare them for further operations like multiplication and combining like terms.

Exponent multiplication occurs when you have terms with the same base that are being multiplied together. The key rule here is \( a^m \times a^n = a^{m+n} \). This is because you simply add the exponents when multiplying. For instance, in the given solution, we combine \( p^3 \) and \( p^{12} \) to get \( p^{3+12} = p^{15} \).
This rule simplifies the equation significantly and reduces the complexity of the expression. Understanding exponent multiplication allows you to handle multiple terms with the same base and simplifies the overall algebraic structure.

Combining like terms is essential for simplifying algebraic expressions. In algebra, like terms are terms that have the same variables raised to the same power. For example \( 3a^2 \) and \( 5a^2 \) are like terms because they both include \( a^2 \).
When you combine like terms, you simply add or subtract their coefficients. In our example, combining \( 8p^3q^{12} \) and \( 25p^{12}q^2 \) involves multiplying their coefficients and exponents separately, leading to \( 200p^{15}q^{14} \). This crucial step makes the expression much easier to manage.

An algebraic expression involves numbers, variables, and arithmetic operations. They can be simplified using fundamental rules like the power rule and operations with exponents. Simplifying algebraic expressions is about making an equation more compact and easier to understand.
In the example exercise, we start with an expression \( \text{(2pq}^{4})^{3}(\text{5p}^{6}\text{q})^{2} \). By applying algebraic rules systematically, we transform it into \( 200p^{15}q^{14} \).
Understanding the components of algebraic expressions and the rules that govern them is crucial for solving more complex algebra problems efficiently. Aim to recognize patterns and apply appropriate rules to simplify and solve algebraic expressions accurately.