Finding The Value Of C In The Equation: A Step-by-Step Solution

#Introduction

In this article, we will delve into the process of finding the specific value of c that satisfies the given equation, ensuring that the equation holds true under the conditions that x and y are both positive values. The equation we aim to solve involves cube roots and fractions, adding an interesting layer of complexity to the problem. Our goal is to provide a step-by-step explanation, making the solution accessible and clear. The equation we are tasked with solving is:

$$\sqrt[3]{\frac{x^3}{c y^4}}=\frac{x}{4 y(\sqrt[3]{y})}$$

This equation involves variables, cube roots, and fractions, requiring a strategic approach to isolate c and determine its value. We will explore each step in detail, from simplifying the equation to isolating c and arriving at the solution. Understanding the underlying principles of algebraic manipulation and root simplification is crucial for successfully tackling this problem. So, let's embark on this mathematical journey to uncover the value of c that makes this equation true.

Breaking Down the Equation: Step-by-Step Solution

Let's embark on a step-by-step journey to solve the equation and find the value of c that makes it true. This involves a methodical approach, carefully simplifying and manipulating the equation until we isolate c. Remember, our initial equation is:

$$\sqrt[3]{\frac{x^3}{c y^4}}=\frac{x}{4 y(\sqrt[3]{y})}$$

Our first step involves simplifying the cube root on the left-hand side of the equation. To do this effectively, we'll apply the properties of radicals and exponents, ensuring each step is clear and concise.

Step 1: Simplifying the Cube Root

To begin, we focus on the left side of the equation: $\sqrt[3]{\frac{x^3}{c y^4}}$. The cube root of a fraction can be expressed as the fraction of the cube roots, allowing us to rewrite the expression as:

$$\frac{\sqrt[3]{x^3}}{\sqrt[3]{c y^4}}$$

Now, we simplify the numerator. The cube root of x cubed, denoted as $\sqrt[3]{x^3}$, is simply x. This is because the cube root operation effectively undoes the cubing operation. So, our expression now looks like this:

$$\frac{x}{\sqrt[3]{c y^4}}$$

Next, we need to address the denominator, $\sqrt[3]{c y^4}$. We can separate the cube root of the product cy⁴ into the product of cube roots. This gives us:

$$\sqrt[3]{c} \cdot \sqrt[3]{y^4}$$

To further simplify $\sqrt[3]{y^4}$, we can rewrite y⁴ as y³ * y. This allows us to take the cube root of y³, which is y, leaving us with a simplified expression. The cube root of y⁴ can then be expressed as:

$$\sqrt[3]{y^4} = \sqrt[3]{y^3 \cdot y} = y \sqrt[3]{y}$$

Substituting this back into our denominator, we get:

$$\sqrt[3]{c} \cdot y \sqrt[3]{y}$$

Therefore, the left-hand side of our original equation, after simplification, becomes:

$$\frac{x}{\sqrt[3]{c} \cdot y \sqrt[3]{y}}$$

This simplification is a crucial step in isolating c. By breaking down the cube root and applying the properties of radicals, we've made the equation more manageable. In the next step, we'll incorporate this simplified expression back into the original equation and continue our journey towards finding the value of c.

Step 2: Incorporating the Simplified Expression

Now that we've successfully simplified the left side of the equation, our next step is to incorporate this simplified expression back into the original equation. This will allow us to progress towards isolating c and ultimately finding its value. Recall that the simplified form of the left side is:

$$\frac{x}{\sqrt[3]{c} \cdot y \sqrt[3]{y}}$$

Our original equation was:

$$\sqrt[3]{\frac{x^3}{c y^4}}=\frac{x}{4 y(\sqrt[3]{y})}$$

Substituting the simplified left side into the equation, we get:

$$\frac{x}{\sqrt[3]{c} \cdot y \sqrt[3]{y}} = \frac{x}{4 y(\sqrt[3]{y})}$$

This substitution is a key step as it brings us closer to isolating c. We now have an equation where both sides share similar terms, making it easier to manipulate and simplify further. The next step involves strategically canceling out common terms to streamline the equation. This will help us to focus specifically on the terms that involve c, making the isolation process more straightforward.

Step 3: Canceling Common Terms

With the simplified equation in hand, we can now proceed to cancel out the common terms present on both sides. This step is crucial for further simplifying the equation and bringing us closer to isolating c. Our equation, after substituting the simplified left side, is:

$$\frac{x}{\sqrt[3]{c} \cdot y \sqrt[3]{y}} = \frac{x}{4 y(\sqrt[3]{y})}$$

Observing the equation, we can identify common factors in the numerators and denominators. Notice that x appears in the numerator on both sides of the equation. Since we are given that x > 0*, we can safely divide both sides by x without affecting the equality. Similarly, the term y appears in the denominator on both sides, and since y > 0*, we can also divide both sides by y. Additionally, the term $\sqrt[3]{y}$ is present in the denominator on both sides, allowing us to divide by this term as well.

By canceling these common terms, we effectively reduce the complexity of the equation, making it easier to work with. After canceling x, y, and $\sqrt[3]{y}$ from both sides, our equation simplifies to:

$$\frac{1}{\sqrt[3]{c}} = \frac{1}{4}$$

This simplified form is significantly easier to manage and directly addresses the term containing c. The next step will involve isolating $\sqrt[3]{c}$ to further our progress in solving for c. This cancellation of common terms is a fundamental technique in algebra, allowing us to streamline equations and focus on the essential components needed to solve for the unknown variable.

Step 4: Isolating the Cube Root of c

After canceling the common terms in the previous step, we have arrived at a much simpler equation. This simplification brings us closer to isolating c and finding its value. Our equation now stands as:

$$\frac{1}{\sqrt[3]{c}} = \frac{1}{4}$$

To isolate $\sqrt[3]{c}$, we can take the reciprocal of both sides of the equation. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. Taking the reciprocal of both sides maintains the equality and helps us to bring $\sqrt[3]{c}$ to the numerator.

Taking the reciprocal of the left side, $\frac{1}{\sqrt[3]{c}}$, gives us $\sqrt[3]{c}$. Similarly, taking the reciprocal of the right side, $\frac{1}{4}$, gives us 4. Therefore, our equation becomes:

$$\sqrt[3]{c} = 4$$

Now, we have successfully isolated the cube root of c. This is a significant step forward, as we are now one step away from finding the value of c itself. The next and final step involves eliminating the cube root to solve for c directly. This isolation of $\sqrt[3]{c}$ demonstrates the power of algebraic manipulation in simplifying equations and making them easier to solve. In the next step, we will employ a straightforward method to remove the cube root and determine the value of c.

Step 5: Solving for c

Having isolated the cube root of c, we are now in the final stage of solving for c. Our equation at this point is:

$$\sqrt[3]{c} = 4$$

To eliminate the cube root and find the value of c, we need to perform the inverse operation of taking a cube root. The inverse operation is cubing, which means raising both sides of the equation to the power of 3. This will effectively undo the cube root on the left side, leaving us with c.

Cubing both sides of the equation, we get:

$$(\sqrt[3]{c})^3 = 4^3$$

The cube root of c, when cubed, simply becomes c. On the right side, 4 cubed, or 4³, is 4 multiplied by itself three times, which equals 64. Therefore, our equation now reads:

$$c = 64$$

Thus, we have successfully found the value of c that satisfies the original equation. By systematically simplifying the equation, isolating the cube root of c, and then cubing both sides, we have determined that c equals 64. This final step completes our journey of solving the equation, demonstrating the effectiveness of algebraic techniques in finding solutions to complex problems.

Conclusion

In conclusion, the value of c that makes the equation $\sqrt[3]{\frac{x^3}{c y^4}}=\frac{x}{4 y(\sqrt[3]{y})}$ true, given that x > 0* and y > 0*, is 64. We arrived at this solution through a methodical step-by-step process, which involved simplifying the cube root, incorporating the simplified expression back into the equation, canceling common terms, isolating the cube root of c, and finally, solving for c by cubing both sides of the equation. This exercise not only provides the answer but also highlights the importance of understanding and applying algebraic principles to solve mathematical problems.

The process of solving this equation demonstrates how complex-looking problems can be broken down into manageable steps. Each step, from simplifying radicals to isolating variables, requires a clear understanding of mathematical operations and properties. By carefully applying these principles, we were able to navigate through the complexities of the equation and arrive at a definitive solution. The value c = 64 is the key that unlocks the truth of the equation, making it a valid statement under the given conditions. This journey through algebra underscores the power and elegance of mathematical problem-solving.

Value of c, equation, cube root, simplification, algebraic manipulation, solving for variables, mathematical problem-solving.