Simplifying Expressions: Solving (5m^-2)(2m^-3)

In the realm of algebra, mastering the manipulation of exponents is crucial for simplifying expressions and solving complex equations. This article delves into the step-by-step process of finding the product of $\\left(5 m^{-2}\\\right)\\left(2 m^{-3}\\\right)$, offering a comprehensive explanation suitable for students and anyone looking to refresh their understanding of exponent rules. We will break down the problem, explore the underlying principles, and provide clear, concise steps to arrive at the solution. This detailed guide will not only help you solve this specific problem but also equip you with the skills to tackle similar algebraic challenges with confidence. Understanding the rules of exponents is fundamental to success in mathematics and related fields, making this a vital concept to grasp. So, let's embark on this mathematical journey together and unlock the secrets of exponent manipulation.

Decoding the Expression: Initial Assessment

Before diving into the calculations, it’s essential to understand the components of the expression $\\\left(5 m^{-2}\\ ight)\\\\left(2 m^{-3}\\ ight)$. The expression involves two terms, each consisting of a coefficient (a numerical value) and a variable (m) raised to a negative exponent. The first term is $5m^{-2}$, where 5 is the coefficient and $m^{-2}$ signifies m raised to the power of -2. The second term is $2m^{-3}$, with 2 as the coefficient and $m^{-3}$ indicating m raised to the power of -3. The task is to find the product of these two terms, which means we need to multiply them together. This requires applying the rules of exponents, particularly the product of powers rule, which states that when multiplying expressions with the same base, you add the exponents. Negative exponents indicate reciprocals, so understanding their implications is crucial for accurate simplification. By carefully assessing each component, we can develop a strategic approach to solving the problem. This initial assessment sets the stage for a clear and efficient solution process, ensuring we address each aspect of the expression accurately.

Step-by-Step Solution: Multiplying Coefficients and Variables

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