Simplifying (5m^-2)(2m^-3) A Step-by-Step Guide

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Mathematics often presents us with expressions that require simplification. One such expression is the product of (5m-2)(2m-3). This article aims to provide a detailed, step-by-step explanation of how to solve this problem, ensuring a clear understanding of the underlying mathematical principles. We'll break down the components, apply the relevant rules of exponents, and arrive at the simplified answer. Whether you're a student grappling with algebra or someone looking to brush up on your math skills, this guide will equip you with the knowledge to tackle similar problems with confidence.

Breaking Down the Expression

To effectively tackle the product of (5m-2)(2m-3), we first need to understand the components of the expression. We have two terms being multiplied: 5m^-2 and 2m^-3. Each term consists of a coefficient (the numerical part) and a variable with an exponent. In the first term, 5 is the coefficient, m is the variable, and -2 is the exponent. Similarly, in the second term, 2 is the coefficient, m is the variable, and -3 is the exponent. Understanding these components is crucial because it allows us to apply the rules of algebra correctly. Recognizing the coefficients and the variables with their respective exponents sets the stage for simplifying the expression by combining like terms and applying the exponent rules. This initial breakdown is a fundamental step in solving any algebraic expression, making the subsequent steps more manageable and less prone to errors. By clearly identifying each part, we can proceed with a methodical approach to simplification.

Applying the Product of Powers Rule

Now that we've broken down the expression (5m-2)(2m-3), the next step is to apply the product of powers rule. This rule is a fundamental concept in algebra and is essential for simplifying expressions involving exponents. The product of powers rule states that when multiplying terms with the same base, you add their exponents. In mathematical terms, this is expressed as a^m * a^n = a^(m+n), where 'a' is the base and 'm' and 'n' are the exponents. In our expression, the base is 'm', and the exponents are -2 and -3. Thus, we will add these exponents together. Before applying the rule, it's helpful to rearrange the expression to group the coefficients and the variables together: (5 * 2) * (m^-2 * m^-3). This rearrangement makes it clearer how to apply both the multiplication of coefficients and the product of powers rule. Understanding and correctly applying this rule is pivotal for simplifying algebraic expressions efficiently and accurately. The product of powers rule is not just a mathematical shortcut; it’s a reflection of the underlying principles of exponentiation, making it a cornerstone of algebraic manipulation.

Multiplying Coefficients and Adding Exponents

With the expression (5m-2)(2m-3) rearranged as (5 * 2) * (m^-2 * m^-3), we are now ready to perform the multiplication of coefficients and the addition of exponents. First, we multiply the coefficients: 5 multiplied by 2 equals 10. This gives us the numerical part of our simplified term. Next, we focus on the variables with exponents. According to the product of powers rule, we add the exponents when multiplying terms with the same base. In this case, we add -2 and -3: -2 + (-3) = -5. This means the exponent of 'm' in our simplified term will be -5. Combining these two results, we get 10m^-5. This step is crucial as it directly applies the core algebraic principles of multiplying coefficients and adding exponents, leading us closer to the final simplified form of the expression. By meticulously following these steps, we ensure accuracy and a thorough understanding of the process involved in simplifying algebraic expressions.

Dealing with Negative Exponents

Having simplified the expression (5m-2)(2m-3) to 10m^-5, the next step involves addressing the negative exponent. In mathematics, negative exponents indicate the reciprocal of the base raised to the positive of that exponent. In simpler terms, a^-n = 1/a^n. Applying this rule to our expression, we need to convert m^-5 into its reciprocal form. This means m^-5 becomes 1/m^5. Now, we can rewrite the expression 10m^-5 as 10 * (1/m^5). This transformation is crucial because it allows us to express the term with a positive exponent, which is often preferred in simplified algebraic expressions. Understanding how to handle negative exponents is a fundamental skill in algebra, as it allows for a more standard and easily interpretable representation of mathematical terms. This step not only simplifies the expression but also makes it more mathematically elegant and conventionally acceptable.

Expressing the Final Simplified Form

After dealing with the negative exponent in 10m^-5, we arrived at 10 * (1/m^5). The final step is to express this in a simplified, standard algebraic form. To do this, we simply multiply the coefficient 10 by the fraction 1/m^5, which gives us 10/m^5. This is the fully simplified form of the original expression (5m-2)(2m-3). Expressing the result as a fraction with a positive exponent is a conventional way to present algebraic solutions, making it easier to understand and work with in further calculations or applications. This final simplification not only provides the answer but also demonstrates a mastery of algebraic manipulation, showcasing the ability to transform a complex expression into its most basic and understandable form. The journey from the initial expression to this final simplified form highlights the power and elegance of algebraic principles in action.

Conclusion

In conclusion, the product of (5m-2)(2m-3), when simplified, is 10/m^5. This simplification process involved several key steps: breaking down the expression, applying the product of powers rule, multiplying coefficients and adding exponents, dealing with negative exponents, and expressing the final simplified form. Each step is rooted in fundamental algebraic principles, demonstrating the importance of understanding and applying these rules correctly. This exercise not only provides a solution to a specific problem but also reinforces the broader concepts of algebraic manipulation and simplification. By mastering these techniques, one can confidently approach a wide range of mathematical expressions and problems, making the journey through algebra more accessible and rewarding. The ability to simplify expressions like this is a cornerstone of mathematical literacy, essential for further studies in mathematics and its applications in various fields.