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The power rule is a fundamental concept in exponents. It simplifies expressions with exponents by distributing the power over each element within parentheses. For example, if you have an expression like \((xy)^n\), the power rule indicates that you multiply the exponent \(n\) to each component inside, which results in \(x^n \times y^n\).

What's great about the power rule is its versatility. It applies equally to numbers, variables, or a mix of both under multiplication. In our case, applying the power rule to \((2ab)^5\) allows us to simplify this expression by evaluating \((2)^5\), \(a^5\), and \(b^5\) separately.

Simplifying expressions involves breaking down an algebraic expression into its simplest form. Often, this means reducing the number of terms or using exponents efficiently. This procedure makes expressions easier to understand and work with.

In this exercise, we simplified \((2ab)^5\) by using the power rule, resulting in \(2^5 \times a^5 \times b^5\). Calculating further, \(2^5 = 32\), so the simplified expression becomes \(32a^5b^5\). Here, simplifying doesn’t mean performing any more calculations on variables \(a\) and \(b\). Simplification is complete when you can no longer break down the expression into smaller parts without further information on the variables.

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It forms the foundation for most mathematics, helping solve equations and simplify expressions.

One of the foundational elements of algebra is understanding how to handle exponents. This includes knowing how to apply rules such as the power rule, as well as how to combine and reduce terms in an expression, as seen in \((2ab)^5\). It's essential to grasp these basics as they are building blocks for more complicated algebraic concepts.

Keeping the fundamentals sharp – like knowing that you distribute the power across terms when dealing with products within exponents – equips students with the skills to simplify expressions effectively. This helps when moving on to solve more complex equations in algebra and even in calculus and beyond.