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Simplifying expressions in algebra means reducing them to their simplest form without changing their value. This process often makes calculations easier and more efficient. In expressions involving exponents, simplification generally includes combining powers and using skills like factoring, distributing, and canceling terms.

For expressions that include the same base raised to different exponents, we can add or multiply the exponents according to basic algebra rules. In this exercise, we have terms like \(r^{3} r^{2}\) and \(r^{3} r^{5}\). By adding the exponents of the same base \(r\), we transform the expression into \((r^{5})^{4} (r^{8})^{2}\) as a simpler form.

Learning to identify and simplify these patterns is crucial in algebra and helps tremendously with solving equations and evaluating expressions.

Algebraic rules are foundational principles that help us manipulate and solve algebraic expressions confidently. These rules guide the simplification and calculation processes in different mathematical operations.

One important rule of exponentiation is the power of a power rule, which states \((a^n)^m = a^{n*m}\). This rule was used in our exercise to transform \((r^{5})^{4}\) into \(r^{20}\) and \((r^{8})^{2}\) into \(r^{16}\). By applying these rules consistently, solving expressions becomes a systematic and organized process.

Memorizing and understanding algebraic rules will enable you to solve even complex algebraic equations efficiently. They are tools that make algebra less intimidating and more practical in everyday calculations.

Combining like terms is a key algebraic technique to simplify expressions further, especially when they involve similar bases. "Like terms" are terms whose variables (and their exponents) match. The process involves adding or subtracting the coefficients as appropriate while keeping the variable part unchanged.

In the context of exponentiation, like terms have the same base, and their exponents can be added. Hence, in our exercise, once we have \(r^{20}\) and \(r^{16}\), we combined these terms into \(r^{36}\) by adding the exponents because they shared the base \(r\).

Remember, combining like terms does not change the essence or the value of the expression—it merely provides a simplified version that is often cleaner and easier to understand and compute.