Simplifying Exponential Expressions Solving $\frac{6ab}{(a^0b^2)^4}$

Jul 15, 2025 by Admin 71 views

$Unraveling Exponential Expressions A Comprehensive Guide to Simplifying ${\frac{6ab}{(a^0b^2)^4}}$$

Exponential expressions can often appear daunting, but with a systematic approach and a firm understanding of the rules of exponents, they can be simplified effectively. In this article, we will dissect the expression ${\frac{6ab}{(a^0b^2)^4}}$ , meticulously applying the laws of exponents to arrive at its most simplified form. This exploration is crucial for students, educators, and anyone involved in mathematical problem-solving, as it reinforces fundamental algebraic concepts and cultivates analytical skills. The process involves several key steps, each relying on specific exponential rules that must be applied in the correct sequence to avoid errors. By the end of this comprehensive guide, you'll not only understand how to simplify this particular expression but also gain a deeper appreciation for the elegance and consistency of mathematical operations.

Understanding the problem statement is the first step in finding the right solution. The problem presents an algebraic expression involving variables, coefficients, and exponents, and it asks us to find an equivalent expression in a simplified form. The given expression is ${\frac{6ab}{(a^0b^2)^4}}$ . Our goal is to reduce this expression to its simplest form by applying the rules of exponents. This involves dealing with powers of products, powers of powers, and the zero exponent, as well as simplifying the resulting fractions. Simplifying such expressions is a fundamental skill in algebra and is crucial for solving more complex equations and problems in higher mathematics. It allows us to manipulate and understand mathematical relationships more clearly and efficiently.

Before we dive into the solution, let's recap the essential rules of exponents that we'll be using. These rules are the building blocks of simplifying exponential expressions, and mastering them is key to success in algebra and beyond. First, the power of a power rule states that ${(a^m)^n = a^{mn}}$ , which means when you raise a power to another power, you multiply the exponents. Second, the power of a product rule, ${(ab)^n = a^n b^n}$ , tells us that the power of a product is the product of the powers. Third, the zero exponent rule, ${a^0 = 1}$ (where ${a \neq 0}$ ), states that any non-zero number raised to the power of zero is one. Finally, the negative exponent rule, ${a^{-n} = \frac{1}{a^n}}$ , indicates that a negative exponent means we take the reciprocal of the base raised to the positive exponent. These rules are not just arbitrary formulas; they are logical extensions of the definition of exponents and are consistent across various mathematical contexts. Understanding why these rules work, rather than just memorizing them, will make you a more proficient problem solver.

Step-by-Step Simplification

1. Applying the Power of a Power Rule

In this step, we'll focus on simplifying the denominator of the given expression. The denominator is ${(a^0b^2)^4}$ . According to the power of a power rule, we need to multiply the exponents inside the parentheses by the exponent outside the parentheses. This rule states that ${(a^m)^n = a^{mn}}$ . Applying this rule to our expression, we get:

(a^0b^2)^4 = a^{0 \times 4} b^{2 \times 4}

This simplifies to:

a^0 b^8

Now, we also need to remember the zero exponent rule, which states that any non-zero number raised to the power of zero is one. That is, ${a^0 = 1}$ . Applying this to our expression, we get:

1 \times b^8 = b^8

So, the denominator simplifies to ${b^8}$ . This step is crucial because it significantly reduces the complexity of the expression, making it easier to manage in subsequent steps. Understanding and correctly applying the power of a power rule, along with the zero exponent rule, is essential for simplifying exponential expressions. These rules are not isolated concepts but are interconnected parts of the broader framework of exponential operations. Mastering them allows for more efficient and accurate algebraic manipulations.

2. Rewriting the Expression

Having simplified the denominator, we can now rewrite the entire expression. The original expression was:

\frac{6ab}{(a^0b^2)^4}

And we've simplified the denominator ${(a^0b^2)^4}$ to ${b^8}$ . So, the expression now becomes:

\frac{6ab}{b^8}

This step is straightforward but important as it brings us closer to the final simplified form. By substituting the simplified denominator back into the original expression, we reduce the complexity of the problem and make it more manageable. This process of breaking down a complex problem into smaller, more manageable parts is a key strategy in problem-solving, not just in mathematics but in many other fields as well. It allows us to focus on specific aspects of the problem and avoid being overwhelmed by the whole. The rewritten expression is now in a form where we can apply the quotient rule of exponents, which will further simplify it.

3. Applying the Quotient Rule

Now, we need to simplify the expression ${\frac{6ab}{b^8}}$ . To do this, we can use the quotient rule of exponents, which states that ${\frac{a^m}{a^n} = a^{m-n}}$ . In our expression, we have ${\frac{b}{b^8}}$ , which can be written as ${\frac{b^1}{b^8}}$ . Applying the quotient rule, we subtract the exponents:

b^{1-8} = b^{-7}

So, ${\frac{b}{b^8}}$ simplifies to ${b^{-7}}$ . Now, let's rewrite the entire expression with this simplification:

6a \times b^{-7}

This step is critical because it utilizes one of the fundamental rules of exponents to further reduce the expression. The quotient rule is particularly useful when dealing with fractions involving exponents, as it provides a direct way to combine terms with the same base. Understanding this rule and how to apply it is essential for simplifying algebraic expressions and solving equations. The result, ${6ab^{-7}}$ , is already simpler than the original expression, but we can take it one step further by dealing with the negative exponent.

4. Handling the Negative Exponent

We have the expression ${6ab^{-7}}$ . To eliminate the negative exponent, we use the rule that ${a^{-n} = \frac{1}{a^n}}$ . Applying this rule to ${b^{-7}}$ , we get:

b^{-7} = \frac{1}{b^7}

Now, we substitute this back into our expression:

6a \times \frac{1}{b^7}

This simplifies to:

\frac{6a}{b^7}

This step is crucial as it transforms the expression into a form without negative exponents, which is often preferred in simplified expressions. Negative exponents can sometimes be confusing, and converting them to positive exponents makes the expression easier to interpret and work with. The rule for handling negative exponents is a direct consequence of the definition of exponents and is a powerful tool in algebraic manipulation. By applying this rule, we've arrived at the final simplified form of the expression.

Conclusion

After systematically applying the rules of exponents, we have successfully simplified the expression ${\frac{6ab}{(a^0b^2)^4}}$ . Let's recap the steps we took:

Power of a Power Rule: Simplified ${(a^0b^2)^4}$ to ${b^8}$ .
Rewriting the Expression: Rewrote the expression as ${\frac{6ab}{b^8}}$ .
Quotient Rule: Simplified ${\frac{b}{b^8}}$ to ${b^{-7}}$ .
Negative Exponent: Eliminated the negative exponent to get the final simplified form.

Therefore, the equivalent expression to ${\frac{6ab}{(a^0b^2)^4}}$ is:

\frac{6a}{b^7}

This matches option B in the given choices. Simplifying exponential expressions is a fundamental skill in algebra, and this detailed walkthrough demonstrates the importance of understanding and applying the rules of exponents correctly. By mastering these rules, you can confidently tackle more complex algebraic problems and gain a deeper appreciation for the elegance and power of mathematical notation. Remember, each step in the simplification process is a logical consequence of the rules, and by understanding the logic behind each rule, you can avoid memorization and develop a more intuitive understanding of algebra. This understanding will serve you well in future mathematical endeavors.

Based on our step-by-step simplification, we arrived at the expression ${\frac{6a}{b^7}}$ , which corresponds to option B. Therefore, the correct answer is B. This process underscores the importance of meticulous application of rules and attention to detail when simplifying expressions. Each step must be justified by a specific rule, and any error in applying these rules can lead to an incorrect result. Moreover, this exercise highlights the significance of understanding the underlying concepts rather than simply memorizing formulas. By truly grasping the rules of exponents, one can approach simplification problems with confidence and accuracy.

Simplifying exponential expressions is a cornerstone of algebraic proficiency. By mastering the rules of exponents and practicing their application, you can develop a strong foundation for more advanced mathematical concepts. This exercise with ${\frac{6ab}{(a^0b^2)^4}}$ serves as a valuable example of how these rules work in concert to simplify complex expressions. Remember to always break down complex problems into smaller, more manageable steps, and meticulously apply the appropriate rules. With practice and a solid understanding of the fundamentals, you can confidently tackle any exponential expression that comes your way. The skills you develop in simplifying expressions will not only help you in mathematics but also in various other fields that rely on analytical and problem-solving abilities.