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The rule of exponents is key to simplifying expressions that involve powers. When you have an expression like \( (ab)^n \), this rule allows us to apply the exponent to each factor within the parenthesis separately, turning \( (ab)^n \) into \( a^n b^n \). This is particularly useful with variables and coefficients, such as in the expression \( (8x^3)^2 \). Here, we treat the numerical coefficient and the variable part individually.

So according to our rule, \( (8x^3)^2 \) simplifies to \( 8^2 \times (x^3)^2 \). What we did here was we raised both the coefficient 8 and the term \( x^3 \) to the second power independently. This rule is a cornerstone when it comes to working with exponents because it paves the way for further simplification steps by treating compound expressions in a distributed manner.

Evaluating exponents is the process of calculating the powers of numbers or expressions. Once we have applied the rule of exponents, we can move forward with evaluating the individual terms. For instance, in our example, we calculate \( 8^2 \) to get 64. It's a straightforward computation involving repeated multiplication of the base, which in this case is 8.

For the variable part, \( x^3 \), raised to the second power, we evaluate the exponent by multiplying the powers, applying the specific rule \( (a^m)^n = a^{m\times n} \). Our variable \( x \), initially raised to the third power, is now raised to the power of 6 (\( 3 \times 2 = 6 \)), resulting in \( x^6 \). When evaluating exponents with variables, remember this multiplication-of-powers rule to quickly find the new exponent.

Algebraic simplification is the process of reducing expressions to their simplest form. After applying the rule of exponents and evaluating the exponents, we combine everything to give the expression in its most condensed and straightforward version. Our example yields 64 for the coefficient and \( x^6 \) for the variable part.

By combining these results, we end up with the simplified expression \( 64x^6 \). The objective of algebraic simplification is to rewrite expressions more compactly, making them easier to read, comprehend, and possibly further manipulate. Keep in mind that simplification not only makes expressions neater but also crucial for solving equations, comparing expressions, and performing other algebraic operations.