Finding Values Of A And B For Equivalent Radical Expressions

In the realm of mathematics, particularly when dealing with radical expressions, the quest for simplification and equivalence often leads us down intriguing paths. This article delves into a specific problem: finding the values of 'a' and 'b' that make two seemingly different radical expressions equivalent. We'll embark on a step-by-step journey, unraveling the intricacies of radical simplification and variable manipulation to arrive at the solution. This exploration is not just about finding the answer; it's about understanding the underlying principles that govern mathematical expressions.

Decoding the Problem: A Deep Dive into Radical Expressions

The heart of our challenge lies in the equation: √((126xy⁵)/(32x³)) = √((63y⁵)/(ax^b)). Our mission is to determine the numerical value of 'a' and 'b' that will uphold this equality, given the constraints that x > 0 and y ≥ 0. These constraints are crucial, as they ensure that we're operating within the domain of real numbers, where square roots of negative numbers are undefined. Let's dissect the given equation to forge our path towards the solution.

The first expression, √((126xy⁵)/(32x³)), appears complex at first glance. However, a closer examination reveals opportunities for simplification. The coefficients, 126 and 32, share common factors, and the variables x and y possess exponents that invite algebraic manipulation. By systematically reducing the fraction under the radical and applying the rules of exponents, we can transform this expression into a more manageable form. This simplification is the first crucial step in our journey, setting the stage for comparing it with the second expression.

The second expression, √((63y⁵)/(ax^b)), presents a different kind of challenge. It contains the unknowns 'a' and 'b', which we are tasked with discovering. This expression is already in a relatively simplified form, suggesting that our strategy should focus on manipulating the first expression to match its structure. By aligning the two expressions, we can establish a direct correspondence between their components, ultimately revealing the values of 'a' and 'b'. This process highlights the importance of strategic problem-solving in mathematics, where choosing the right approach can significantly streamline the solution.

Simplifying the First Expression: A Step-by-Step Approach

Our initial task is to simplify the first expression, √((126xy⁵)/(32x³)). To do this, we'll break down the process into manageable steps, focusing on reducing the fraction inside the square root and applying the rules of exponents. This systematic approach will not only lead us to a simplified form but also reinforce our understanding of algebraic manipulation.

1. Reducing the Coefficients: The fraction 126/32 can be simplified by finding the greatest common divisor (GCD) of 126 and 32. The GCD is 2, so we divide both the numerator and the denominator by 2, resulting in 63/16. This step immediately makes the expression less cumbersome.
2. Simplifying the Variables: We now turn our attention to the variables x and y. The expression contains x in both the numerator and the denominator, with exponents of 1 and 3, respectively. Applying the rule of exponents, which states that x^m / x^n = x^(m-n), we can simplify the x terms. In this case, x¹ / x³ = x^(1-3) = x⁻². This means that x effectively moves to the denominator, becoming x². The variable y, with an exponent of 5 in the numerator, remains untouched for now.
3. Rewriting the Expression: After these simplifications, our expression now looks like this: √((63y⁵)/(16x²)). This form is significantly cleaner than the original, making it easier to compare with the second expression. We've successfully reduced the coefficients and simplified the variable terms, bringing us closer to our goal.

Manipulating the Simplified Expression: Isolating the Components

Now that we've simplified the first expression to √((63y⁵)/(16x²)), our next step is to manipulate it further to match the form of the second expression, √((63y⁵)/(ax^b)). This involves isolating the components and making them directly comparable. This manipulation is a crucial step in solving for 'a' and 'b'.

1. Separating the Square Root: We can separate the square root of a fraction into the fraction of the square roots. This gives us √(63y⁵) / √(16x²). This separation allows us to deal with the numerator and the denominator independently, making the simplification process more manageable.
2. Simplifying the Denominator: The denominator, √(16x²), is a perfect square. The square root of 16 is 4, and the square root of x² is x (since x > 0). Therefore, the denominator simplifies to 4x. This simplification is significant, as it eliminates the radical from the denominator.
3. Rewriting the Expression: Our expression now stands as √(63y⁵) / (4x). This form is even closer to the structure of the second expression. The numerator contains the constant 63 and the variable y, while the denominator contains a constant and the variable x. The next step is to compare this with the second expression to determine the values of 'a' and 'b'.

Comparing Expressions: Unveiling the Values of a and b

With the first expression simplified to √(63y⁵) / (4x) and the second expression given as √((63y⁵)/(ax^b)), we are now at the crucial stage of comparing the two to determine the values of 'a' and 'b'. This comparison involves carefully examining the components of each expression and identifying the corresponding parts.

1. Matching the Numerators: The numerators of both expressions contain the term √(63y⁵). This confirms that the variable y and its exponent are consistent between the two expressions. This match is a positive sign, indicating that our simplification process has been accurate.
2. Focusing on the Denominators: The key to unlocking the values of 'a' and 'b' lies in the denominators. In the simplified first expression, the denominator is 4x. In the second expression, the denominator inside the square root is ax^b. To make the two expressions equivalent, the denominators under the square root must match. This means we need to relate 4x to √(ax^b).
3. Squaring the Denominator: To compare the denominators directly, we need to square the denominator of the simplified first expression. Squaring 4x gives us (4x)² = 16x². Now we can directly compare this with the denominator inside the square root of the second expression, which is ax^b.
4. Equating the Denominators: For the two expressions to be equivalent, we must have 16x² = ax^b. This equation provides us with the direct link to determine the values of 'a' and 'b'. By matching the coefficients and the exponents of x, we can solve for our unknowns.

Solving for a and b: The Final Revelation

We've arrived at the equation 16x² = ax^b, the key to unlocking the values of 'a' and 'b'. This equation is a direct result of comparing the simplified forms of the two original radical expressions. Now, it's a matter of carefully analyzing the equation and extracting the values of our unknowns.

1. Matching the Coefficients: On the left side of the equation, the coefficient of x² is 16. On the right side, the coefficient is 'a'. For the equation to hold true, the coefficients must be equal. Therefore, we can confidently conclude that a = 16. This is our first breakthrough, revealing the numerical value of 'a'.
2. Matching the Exponents: Next, we turn our attention to the exponents of x. On the left side of the equation, the exponent of x is 2. On the right side, the exponent is 'b'. Again, for the equation to hold true, the exponents must be equal. This leads us to the conclusion that b = 2. This is our second and final piece of the puzzle, revealing the value of 'b'.
3. The Solution: We have successfully determined the values of 'a' and 'b' that make the two radical expressions equivalent. Our solution is a = 16 and b = 2. This completes our journey, from the initial complex expressions to the final, elegant solution. The process highlights the power of algebraic manipulation and strategic problem-solving in mathematics.

Conclusion: A Triumph of Algebraic Mastery

In this mathematical expedition, we've successfully navigated the complexities of radical expressions, simplified fractions, manipulated exponents, and ultimately unveiled the values of 'a' and 'b' that ensure the equivalence of two seemingly disparate expressions. The journey itself underscores the importance of a systematic approach to problem-solving, the power of algebraic techniques, and the beauty of mathematical relationships.

We began with the equation √((126xy⁵)/(32x³)) = √((63y⁵)/(ax^b)) and, through a series of carefully executed steps, arrived at the solution a = 16 and b = 2. This solution not only answers the specific question posed but also reinforces our understanding of fundamental mathematical principles.

The process involved simplifying the initial expression by reducing coefficients and applying the rules of exponents. We then separated the square root, simplified the denominator, and strategically manipulated the expression to match the form of the second expression. Finally, by comparing the components of the simplified expressions, we were able to equate the coefficients and exponents, leading us to the values of 'a' and 'b'.

This exercise serves as a testament to the elegance and power of mathematics. By mastering algebraic techniques and cultivating a strategic mindset, we can unravel complex problems and reveal the underlying order and harmony of the mathematical world. The journey of finding 'a' and 'b' is not just about the answer; it's about the skills and insights gained along the way.