Navigating the realm of algebraic equations can sometimes feel like traversing a complex maze, but with the right strategies and a clear understanding of the fundamental principles, even the most daunting equations can be solved with confidence. In this comprehensive guide, we will delve into the step-by-step process of solving the equation $1+\sqrt[3]{5x}=3$, unraveling the mysteries behind radical equations and equipping you with the skills to tackle similar mathematical challenges.
Our primary goal is to solve the equation $1+\sqrt[3]{5x}=3$, and to achieve this, we will employ a systematic approach that involves isolating the radical term, eliminating the radical, and ultimately determining the value of the unknown variable, x. This journey will not only provide you with the solution to this specific equation but also enhance your problem-solving abilities in the broader context of algebra. So, let's embark on this mathematical adventure together and discover the elegant solution that lies within.
1. Isolate the Radical: Unveiling the Heart of the Equation
When confronted with an equation that harbors a radical expression, the initial step towards unraveling the solution is to isolate the radical. This strategic maneuver involves rearranging the equation to position the radical term on one side, effectively separating it from the other terms. By isolating the radical, we create a clearer pathway for eliminating it and progressing towards the solution. In the context of our equation, $1+\sqrt[3]{5x}=3$, the radical term is $\sqrt[3]{5x}$, and our mission is to liberate it from the clutches of the constant term, 1.
To accomplish this isolation, we employ the fundamental principle of algebraic manipulation: performing the same operation on both sides of the equation to maintain equilibrium. In this instance, we subtract 1 from both sides of the equation, effectively neutralizing the +1 on the left side and transposing it to the right side. This subtraction operation yields the following transformation:
Simplifying both sides of the equation, we arrive at the crucial step of radical isolation:
With the radical term, $\sqrt[3]{5x}$, now standing alone on one side of the equation, we have successfully laid the foundation for the next phase of our problem-solving journey. This isolation not only simplifies the equation visually but also allows us to focus our attention specifically on the radical term, paving the way for its eventual elimination. The ability to isolate radicals is a cornerstone of solving radical equations, and mastering this technique will undoubtedly prove invaluable in your mathematical endeavors.
2. Eliminate the Radical: Raising to the Power of the Index
Having successfully isolated the radical term, $\sqrt[3]{5x}$, our next strategic move is to eliminate the radical altogether. This pivotal step involves leveraging the inherent relationship between radicals and exponents. To eradicate a radical, we raise both sides of the equation to the same power as the index of the radical. The index, in this context, signifies the root being taken – in our case, the cube root, denoted by the index 3. By raising both sides of the equation to the power of 3, we effectively undo the cube root operation, paving the way for a more manageable equation.
In our equation, $\sqrt[3]{5x} = 2$, the radical is a cube root, so we must raise both sides to the power of 3. This operation is expressed mathematically as follows:
On the left side of the equation, raising the cube root of $5x$ to the power of 3 effectively cancels out the radical, leaving us with the radicand, $5x$. On the right side, we evaluate 2 raised to the power of 3, which is 2 multiplied by itself three times: 2 * 2 * 2 = 8. Applying these simplifications, our equation transforms into:
With the radical vanquished, we are now left with a straightforward linear equation, a familiar territory in the realm of algebra. This transformation marks a significant milestone in our problem-solving journey, as we have effectively converted a radical equation into a simpler form that can be readily solved using basic algebraic techniques. The ability to eliminate radicals by raising both sides of the equation to the appropriate power is a fundamental skill in solving radical equations, and mastering this technique will empower you to tackle a wide array of mathematical challenges.
3. Solve for x: Unveiling the Solution
With the radical successfully eliminated, we now stand at the threshold of the final stage: solving for x. Our equation, simplified to $5x = 8$, presents a clear path forward. To isolate x and reveal its value, we must undo the multiplication by 5. This is achieved by dividing both sides of the equation by 5, a fundamental algebraic operation that maintains the balance of the equation while isolating the variable of interest.
Performing this division, we obtain:
On the left side, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with x standing alone. On the right side, we have the fraction $\frac{8}{5}$, which represents the quotient of 8 divided by 5. Thus, our equation transforms into:
We have now successfully isolated x and determined its value: $\frac{8}{5}$. This fraction represents the solution to our original equation, $1+\sqrt[3]{5x}=3$. To express this solution in decimal form, we perform the division, yielding approximately 1.6. Therefore, the solution to the equation is x = $\frac{8}{5}$ or 1.6. This value of x satisfies the original equation, meaning that when we substitute $\frac{8}{5}$ for x in the equation $1+\sqrt[3]{5x}=3$, both sides of the equation will be equal. The journey of solving this radical equation has culminated in the unveiling of the solution, a testament to the power of systematic algebraic manipulation.
4. Verify the Solution: Ensuring Accuracy and Validity
Having diligently navigated the steps to arrive at a potential solution, it is paramount to verify the solution. This crucial step serves as a safeguard against extraneous solutions, those deceptive values that emerge during the solving process but do not genuinely satisfy the original equation. To verify our solution, x = $\frac{8}{5}$, we substitute this value back into the original equation, $1+\sqrt[3]{5x}=3$, and meticulously evaluate both sides.
Substituting x = $\frac{8}{5}$ into the original equation, we obtain:
Simplifying the expression inside the cube root, we have:
The cube root of 8 is 2, as 2 * 2 * 2 = 8. Therefore, our equation becomes:
Evaluating the left side, we find that 1 + 2 = 3, which is indeed equal to the right side of the equation. This equality confirms that our solution, x = $\frac{8}{5}$, is not an extraneous solution but a genuine solution that satisfies the original equation. The verification process underscores the importance of not only finding a potential solution but also ensuring its validity within the context of the original equation. This step provides assurance that our algebraic manipulations have led us to the correct answer, reinforcing the integrity of our problem-solving approach.
In conclusion, we have successfully navigated the intricate path of solving the equation $1+\sqrt[3]{5x}=3$, employing a systematic approach that involved isolating the radical, eliminating the radical, solving for x, and verifying the solution. The solution to the equation is x = $\frac{8}{5}$ or 1.6. This journey has not only provided us with the solution to this specific equation but has also equipped us with valuable skills and insights into the world of radical equations. Mastering these techniques will undoubtedly empower you to tackle a wide array of mathematical challenges with confidence and precision. Remember, the key to success in mathematics lies in a clear understanding of the fundamental principles, a systematic approach, and the unwavering dedication to verify your solutions.
