Solving Cube Root Equations Step By Step Solution For √[3]{-7l+4} = 8

This article provides a detailed walkthrough on solving the cube root equation $\sqrt[3]{-7l + 4} = 8$. We will break down each step, ensuring a clear understanding of the process. If you're struggling with solving algebraic equations, particularly those involving cube roots, this guide is for you. Let's dive in!

Understanding Cube Root Equations

Before we tackle the specific problem, it's essential to understand what cube root equations are and how they differ from other types of equations. A cube root equation is an equation where the variable is under a cube root. The cube root of a number x is a value that, when multiplied by itself three times, equals x. For example, the cube root of 8 is 2 because $2 \times 2 \times 2 = 8$. Unlike square roots, cube roots can handle negative numbers because a negative number multiplied by itself three times results in a negative number (e.g., the cube root of -8 is -2).

Cube root equations are solved by isolating the cube root term and then cubing both sides of the equation to eliminate the cube root. This process is the inverse operation of taking the cube root, making it a straightforward method for solving these types of equations. Understanding this inverse relationship is crucial for solving algebraic expressions and mathematical problems involving radicals. The goal is always to simplify the equation to a form where the variable can be easily isolated and solved. Recognizing the properties of cube roots and their interactions with negative numbers will help avoid common mistakes and lead to correct solutions.

Step 1: Isolating the Cube Root

In our equation, $\sqrt[3]{-7l + 4} = 8$, the cube root term is already isolated on the left side of the equation. This is a crucial first step in solving equations, as it sets the stage for eliminating the radical. When dealing with more complex equations, you might need to perform algebraic manipulations to isolate the cube root. This could involve adding, subtracting, multiplying, or dividing terms on both sides of the equation. However, in our case, we can proceed directly to the next step since $\sqrt[3]{-7l + 4}$ is already by itself.

Ensuring the cube root is isolated simplifies the algebraic process significantly. This step is fundamental because it allows us to apply the inverse operation—cubing—to both sides of the equation. Without isolating the cube root, cubing the entire side of the equation would result in a much more complicated expression, making it harder to solve for the variable. Therefore, always double-check this step to ensure you're simplifying the equation as much as possible before moving forward. This methodical approach is vital in mathematical problem-solving, particularly when dealing with radicals and exponents. Remember, a clear and organized approach can make even complex problems manageable. By properly isolating the cube root, we pave the way for a straightforward solution.

Step 2: Cubing Both Sides

To eliminate the cube root, we need to perform the inverse operation, which is cubing. We cube both sides of the equation $\sqrt[3]{-7l + 4} = 8$. This means we raise both sides to the power of 3. Mathematically, this is represented as $(\sqrt[3]{-7l + 4})^3 = 8^3$. Cubing the left side cancels out the cube root, leaving us with the expression inside the cube root, which is $-7l + 4$. On the right side, $8^3$ means 8 multiplied by itself three times, which equals 512.

This step is a fundamental technique in solving algebraic equations involving radicals. The principle behind it is to apply an operation that reverses the radical, effectively simplifying the equation. By cubing both sides, we transform the equation from one involving a cube root to a simpler linear equation. This is a crucial step in mathematical problem-solving, allowing us to isolate the variable and eventually find its value. The resulting equation, $-7l + 4 = 512$, is much easier to solve than the original equation. Understanding the inverse relationship between cube roots and cubing is essential for mastering these types of algebraic manipulations. This step highlights the power of using inverse operations to simplify equations and make them solvable.

Step 3: Simplifying the Equation

After cubing both sides, our equation is now $-7l + 4 = 512$. To isolate the term with the variable, we need to subtract 4 from both sides of the equation. This gives us $-7l = 512 - 4$, which simplifies to $-7l = 508$. This step is a basic but essential part of solving equations, as it helps us move closer to isolating the variable l. Subtracting the same value from both sides maintains the equality of the equation, a fundamental principle in algebra.

Simplifying the equation at each step is crucial in algebraic problem-solving. It breaks down the problem into smaller, more manageable parts, making it easier to track progress and avoid errors. By subtracting 4 from both sides, we've successfully isolated the term containing the variable on one side and a constant on the other. This is a key technique in algebraic manipulations and prepares us for the final step of solving for l. The goal is always to simplify the equation as much as possible before solving for the variable. This methodical approach is a hallmark of effective mathematical strategies and ensures accuracy in the final solution. With the equation now simplified to $-7l = 508$, we are well-positioned to find the value of l.

Step 4: Solving for 'l'

To solve for l in the equation $-7l = 508$, we need to divide both sides by -7. This gives us $l = \frac{508}{-7}$. The fraction $\frac{508}{-7}$ can be simplified to $l = -\frac{508}{7}$. Now, we need to check if this fraction can be further reduced. To do this, we look for common factors between 508 and 7. Since 7 is a prime number, we only need to check if 508 is divisible by 7. Dividing 508 by 7, we get approximately 72.57, which is not an integer, so 508 is not divisible by 7.

This step is crucial in solving algebraic problems as it directly leads to the value of the variable. Dividing both sides by the coefficient of l isolates l, giving us the solution. Simplifying the resulting fraction is important to ensure the answer is in its most reduced form. In this case, we found that the fraction $- \frac{508}{7}$ cannot be reduced further because 508 and 7 have no common factors other than 1. This process of simplification is a key skill in mathematical problem-solving, ensuring that the final answer is presented in the most concise and accurate manner. The ability to identify and eliminate common factors is essential for simplifying fractions and expressing solutions in their simplest form. Thus, the solution to the equation is $l = -\frac{508}{7}$, a reduced fraction that represents the exact value of l.

Final Answer

Therefore, the solution to the equation $\sqrt[3]{-7l + 4} = 8$ is $l = -\frac{508}{7}$. This fraction is already in its reduced form, and it represents the exact value of l that satisfies the given equation. We have successfully solved the algebraic equation by isolating the cube root, cubing both sides, simplifying the equation, and solving for the variable. This step-by-step approach is essential in mathematical problem-solving, ensuring accuracy and clarity in the solution process.

In summary, we began with a cube root equation, applied the inverse operation to eliminate the radical, and then used basic algebraic techniques to isolate and solve for l. The final answer, $l = -\frac{508}{7}$, is a testament to the effectiveness of these methods. Understanding and applying these techniques is crucial for solving a wide range of algebraic problems, particularly those involving radicals and exponents. This detailed solution demonstrates the importance of methodical steps and accurate calculations in arriving at the correct answer.

Conclusion

Solving cube root equations involves a systematic approach that combines algebraic manipulation with an understanding of inverse operations. By isolating the cube root, cubing both sides, simplifying the equation, and solving for the variable, we can find the solution. In this case, the solution to $\sqrt[3]{-7l + 4} = 8$ is $l = -\frac{508}{7}$. This process highlights the fundamental principles of algebraic problem-solving, emphasizing the importance of accuracy, simplification, and a step-by-step methodology.

Mastering these techniques is essential for anyone studying algebra or mathematics in general. The ability to solve cube root equations is a valuable skill that can be applied to more complex problems in various fields, including science and engineering. The methodical approach demonstrated here can be used to tackle a wide range of mathematical challenges, making the understanding of these concepts crucial for academic and professional success. By following these steps and practicing regularly, you can confidently solve cube root equations and enhance your overall algebraic skills.