Marina's Simplified Expression Exponent Of B Explained

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In this article, we delve into the intricacies of simplifying algebraic expressions, focusing on Marina's approach to the expression −4a−2b48a−6b−3\frac{-4 a^{-2} b^4}{8 a^{-6} b^{-3}}. Marina correctly simplified this expression, and we aim to understand the steps involved and the underlying mathematical principles. Our goal is to provide a comprehensive explanation that not only clarifies the solution but also enhances your understanding of exponent rules and algebraic manipulations. We will dissect the original expression, break down each simplification step, and ultimately determine the correct exponent of the variable b in Marina's simplified form. This exploration will serve as a valuable exercise in mastering algebraic simplification techniques.

Understanding the Problem

The problem presents us with an algebraic expression involving negative exponents and fractions. Marina has simplified this expression to the form −12a4b□-\frac{1}{2} a^4 b^{\square}, where our task is to find the missing exponent of b. To solve this, we need to apply the rules of exponents and simplify the original expression step by step. The initial expression is:

−4a−2b48a−6b−3\frac{-4 a^{-2} b^4}{8 a^{-6} b^{-3}}

This expression involves several components: numerical coefficients (-4 and 8), variable a raised to negative exponents (-2 and -6), and variable b raised to exponents 4 and -3. Simplifying this requires us to address each component individually and then combine the results. The key to success lies in understanding how negative exponents work and how to divide terms with the same base. This problem not only tests our ability to apply exponent rules but also our understanding of fraction simplification and algebraic manipulation. By carefully working through each step, we can confidently arrive at the correct exponent for b.

Step-by-Step Simplification

To accurately determine the exponent of b, let's embark on a detailed, step-by-step simplification of the given algebraic expression. This methodical approach will not only unveil the solution but also reinforce our understanding of the fundamental principles governing algebraic manipulations and exponent rules. By dissecting each component of the expression and applying the appropriate rules, we can confidently navigate through the simplification process and arrive at the correct answer.

1. Simplifying the Numerical Coefficients

The first step in simplifying the expression −4a−2b48a−6b−3\frac{-4 a^{-2} b^4}{8 a^{-6} b^{-3}} involves dealing with the numerical coefficients, -4 and 8. We can simplify the fraction formed by these coefficients:

−48=−12\frac{-4}{8} = -\frac{1}{2}

This simplification is a straightforward arithmetic operation. Dividing -4 by 8 results in -1/2, which is the simplified numerical coefficient. This step sets the stage for the subsequent simplification of the variable terms. By addressing the numerical coefficients first, we reduce the complexity of the overall expression and make it easier to focus on the variables and their exponents. This meticulous approach is crucial in algebraic simplification, ensuring that each component is handled correctly and efficiently. The simplified numerical coefficient, -1/2, will be a key part of the final simplified expression.

2. Simplifying the Variable a Terms

Next, we focus on the terms involving the variable a. We have a−2a^{-2} in the numerator and a−6a^{-6} in the denominator. To simplify these terms, we use the rule for dividing exponents with the same base:

aman=am−n\frac{a^m}{a^n} = a^{m-n}

Applying this rule to our expression, we get:

a−2a−6=a−2−(−6)=a−2+6=a4\frac{a^{-2}}{a^{-6}} = a^{-2 - (-6)} = a^{-2 + 6} = a^4

This step demonstrates the crucial role of understanding and applying exponent rules. Subtracting a negative exponent is equivalent to adding the positive value, which is a common area for errors. The simplification of the a terms to a4a^4 is a significant step towards the final solution. It showcases how negative exponents can be manipulated to obtain a positive exponent, making the expression more manageable and easier to interpret. This process not only simplifies the expression but also highlights the importance of precision in mathematical operations.

3. Simplifying the Variable b Terms

Now, let's simplify the terms involving the variable b. We have b4b^4 in the numerator and b−3b^{-3} in the denominator. Again, we apply the rule for dividing exponents with the same base:

b4b−3=b4−(−3)=b4+3=b7\frac{b^4}{b^{-3}} = b^{4 - (-3)} = b^{4 + 3} = b^7

Similar to the simplification of the a terms, this step involves subtracting a negative exponent. The result is b7b^7, which indicates that the exponent of b in the simplified expression is 7. This simplification is a critical part of the solution, as it directly answers the question posed in the problem. The correct application of the exponent rule is paramount here, and the resulting b7b^7 term demonstrates a clear understanding of how to handle negative exponents in division.

4. Combining the Simplified Terms

Finally, we combine the simplified numerical coefficient and the simplified variable terms to obtain the complete simplified expression:

−12a4b7- \frac{1}{2} a^4 b^7

This expression represents the simplified form of the original expression. It incorporates the simplified numerical coefficient (-1/2), the simplified a term (a4a^4), and the simplified b term (b7b^7). The combination of these terms demonstrates the culmination of all the previous simplification steps. This final expression is not only mathematically correct but also clearly presents the relationship between the variables and their exponents. It serves as a testament to the power of algebraic simplification and the importance of a systematic approach.

Determining the Exponent of b

From our step-by-step simplification, we have arrived at the expression −12a4b7-\frac{1}{2} a^4 b^7. Comparing this to Marina's simplified expression, −12a4b□-\frac{1}{2} a^4 b^{\square}, we can clearly see that the missing exponent of b is 7. This conclusion is a direct result of the careful and methodical simplification process we undertook. By applying the rules of exponents and simplifying each component of the expression, we were able to isolate the exponent of b and determine its value. This exercise not only provides the answer to the problem but also reinforces the importance of precision and attention to detail in algebraic manipulations.

Conclusion

In conclusion, by meticulously simplifying the expression −4a−2b48a−6b−3\frac{-4 a^{-2} b^4}{8 a^{-6} b^{-3}}, we have determined that the exponent of the variable b in Marina's simplified expression is 7. This process involved simplifying the numerical coefficients, applying exponent rules to the variable terms, and combining the results. The key takeaway is the importance of understanding and correctly applying the rules of exponents, particularly when dealing with negative exponents. This problem serves as a valuable exercise in algebraic simplification and reinforces the skills necessary for success in more advanced mathematical concepts.

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