Marina's Algebraic Simplification Correcting The Exponent Of B

In this article, we delve into the world of algebraic expressions and explore Marina's attempt to simplify the fraction (-4a^-2b⁴) / (8a^-6b^-3). Our focus is on understanding the steps involved in simplifying such expressions and, more specifically, on determining the correct exponent for the variable b in Marina's final answer. Marina's simplified expression is given as -1/2 a⁴b^□, where our mission is to fill in the blank and uncover the correct exponent. To achieve this, we will meticulously walk through the simplification process, leveraging the fundamental rules of exponents and algebraic manipulation. This exploration is crucial not only for solving this specific problem but also for building a solid foundation in algebra, a skill that's essential for more advanced mathematical concepts. By carefully examining each step and understanding the reasoning behind it, we can confidently arrive at the accurate exponent for b and gain a deeper appreciation for the elegance and power of algebraic simplification.

H2: Understanding the Initial Expression

The algebraic expression we are tasked with simplifying is (-4a^-2b⁴) / (8a^-6b^-3). This expression involves negative exponents, which might seem daunting at first. However, the key to tackling such expressions lies in understanding the fundamental rules of exponents. Recall that a negative exponent signifies the reciprocal of the base raised to the positive value of the exponent. In other words, x^-n = 1/xⁿ. This crucial rule will be our guide as we navigate the simplification process. Furthermore, we have fractional coefficients and multiple variables, each with its own exponent. This complexity underscores the importance of a systematic approach. We need to address each component of the expression individually, applying the relevant rules of exponents and arithmetic operations. By breaking down the expression into smaller, manageable parts, we can avoid confusion and ensure accuracy. The goal is to rewrite the expression in its simplest form, where each variable has a positive exponent and the numerical coefficient is reduced to its lowest terms. This involves careful manipulation and a thorough understanding of the properties of exponents and fractions.

H3: Breaking Down the Expression Step-by-Step

To effectively simplify the expression (-4a^-2b⁴) / (8a^-6b^-3), we'll dissect it into manageable parts, starting with the numerical coefficients. The fraction formed by the coefficients is -4/8, which simplifies to -1/2. This is a straightforward arithmetic operation, but it's a crucial first step. Next, we turn our attention to the variables and their exponents. We have a raised to different powers in the numerator and the denominator: a^-2 in the numerator and a^-6 in the denominator. To simplify this, we use the rule of exponents that states x^m / xⁿ = x^(m-n). Applying this rule, we get a^{-2 - (-6)} = a⁴. This shows how negative exponents can be manipulated to yield positive exponents in the simplified expression. Similarly, for the variable b, we have b⁴ in the numerator and b^-3 in the denominator. Applying the same rule of exponents, we get b^{4 - (-3)} = b⁷. The key takeaway here is the careful handling of negative signs, which is essential for avoiding errors. By systematically addressing each component of the expression – the numerical coefficients and each variable with its exponent – we pave the way for the final simplified form. This step-by-step approach ensures that no detail is overlooked and that the simplification process is both accurate and transparent.

H3: Applying the Quotient of Powers Rule

The quotient of powers rule is a cornerstone of simplifying expressions with exponents. This rule, mathematically expressed as x^m / xⁿ = x^(m-n), provides a direct method for handling division when the bases are the same. In the context of Marina's expression, (-4a^-2b⁴) / (8a^-6b^-3), this rule is crucial for simplifying the terms involving a and b. Let's revisit the a terms: we have a^-2 divided by a^-6. Applying the quotient of powers rule, we subtract the exponents: -2 - (-6). Remember that subtracting a negative number is equivalent to adding its positive counterpart, so we have -2 + 6, which equals 4. Therefore, the simplified term for a is a⁴. This demonstrates the power of the rule in transforming negative exponents into positive ones, a key step in simplification. Now, let's apply the same principle to the b terms: b⁴ divided by b^-3. Again, we subtract the exponents: 4 - (-3), which is 4 + 3, resulting in 7. So, the simplified term for b is b⁷. Understanding and applying the quotient of powers rule correctly is fundamental to simplifying algebraic expressions. It allows us to efficiently combine terms with the same base, regardless of whether the exponents are positive or negative, leading to a more concise and manageable form of the expression. This rule is not just a mathematical trick; it's a reflection of the fundamental properties of exponents and how they interact with division.

H2: Determining the Exponent of b

Having meticulously simplified the expression (-4a^-2b⁴) / (8a^-6b^-3), we now focus on pinpointing the correct exponent for b in Marina's solution. As we worked through the simplification, we applied the quotient of powers rule to the b terms, which were b⁴ in the numerator and b^-3 in the denominator. The rule dictates that we subtract the exponents: 4 - (-3). This subtraction of a negative number translates to addition: 4 + 3. The result is 7. Therefore, the correct exponent for b in the simplified expression is 7. Marina's simplified expression is given as -1/2 a⁴b^□, and we have definitively determined that the missing exponent, represented by the square, should be 7. This result highlights the importance of careful application of the rules of exponents, especially when dealing with negative signs. A minor error in arithmetic can lead to an incorrect exponent and a flawed final answer. By systematically breaking down the expression and applying each rule with precision, we can confidently arrive at the correct solution. The exponent of 7 for b completes the simplified expression, providing a clear and accurate representation of the original fraction.

H3: The Correct Exponent for b is 7

In summary, after carefully applying the quotient of powers rule and simplifying the expression (-4a^-2b⁴) / (8a^-6b^-3), we have conclusively found that the correct exponent for the variable b is 7. This means that Marina's simplified expression, -1/2 a⁴b^□, should have 7 in the box, making it -1/2 a⁴b⁷. Our step-by-step analysis, starting from understanding the initial expression, breaking it down into components, and applying the appropriate rules of exponents, has led us to this precise answer. We began by simplifying the numerical coefficients, then tackled the a terms using the rule x^m / xⁿ = x^(m-n), and finally, we focused on the b terms using the same rule. The subtraction of the negative exponent in the denominator, 4 - (-3), was a crucial step, resulting in the exponent 7 for b. This process demonstrates the importance of attention to detail and a solid grasp of exponent rules in algebraic simplification. The exponent of 7 not only completes the solution but also solidifies our understanding of how variables and their powers interact within algebraic expressions. It's a testament to the power of systematic problem-solving and the beauty of mathematical precision.

H2: Conclusion: Marina's Simplified Expression and the Power of Exponents

In conclusion, our journey through the simplification of the algebraic expression (-4a^-2b⁴) / (8a^-6b^-3) has culminated in the definitive determination of the exponent for the variable b. Through a meticulous step-by-step process, we applied the fundamental rules of exponents, particularly the quotient of powers rule, to arrive at the simplified form. Marina's simplified expression, -1/2 a⁴b^□, now stands complete with the exponent 7 filling the blank, resulting in -1/2 a⁴b⁷. This exercise underscores the critical role that exponents play in algebraic expressions and the importance of mastering their properties. The ability to manipulate exponents, especially negative exponents and fractions, is essential for success in algebra and beyond. Furthermore, this problem highlights the significance of a systematic approach to problem-solving. By breaking down the expression into smaller, manageable parts, we were able to address each component individually and avoid the pitfalls of rushing or overlooking details. The correct application of the quotient of powers rule, coupled with careful attention to arithmetic, ensured an accurate and confident solution. The final simplified expression not only provides the answer to the problem but also reinforces our understanding of the elegance and efficiency of algebraic manipulation. The power of exponents, when harnessed correctly, allows us to express complex relationships in a concise and meaningful way.