Simplify each of the following expressions as completely as...

Simplify each of the following expressions as completely as possible. Final answers should be expressed with positive exponents only. (Assume that all variables represent positive quantities.)$$\left(a^{5} b^{-2}\right)^{-3}$$

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Key Concepts

Power of a Product Rule

This rule states that when a product of factors is raised to an exponent, each factor is raised to that exponent separately. It allows us to simplify expressions like (xy)^n into x^n * y^n, which is a fundamental step in breaking down and understanding more complex exponential expressions.

Power of a Power Rule

The power of a power rule allows us to simplify expressions where an exponential expression is raised to another power. It states that (x^m)^n equals x^(m*n). This rule is essential for reducing nested exponents into a single exponent, thereby simplifying the expression.

Negative Exponent Rule

The negative exponent rule provides a method to rewrite expressions with negative exponents as reciprocals with positive exponents. Specifically, x^(-n) is equivalent to 1/(x^n). This transformation is crucial when the goal is to express the final answer with only positive exponents.

Simplification to Positive Exponents

In many algebraic problems, especially those involving variables assumed to be positive, the final expression is required to have only positive exponents. This involves applying properties of exponents, such as the negative exponent rule, to ensure that all components of the expression are written with positive exponents, making the result easier to interpret and work with.

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