Solving Cube Root Equations A Step By Step Guide

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In this article, we will delve into the process of solving the equation (18−x)3=−12\sqrt[3]{\left(\frac{1}{8}-x\right)}=-\frac{1}{2}. This equation involves a cube root, and our goal is to isolate the variable x and find its value. We will break down the steps involved in solving this equation, providing a clear and comprehensive explanation for each stage. Let's embark on this mathematical journey together!

Understanding Cube Roots

Before we dive into solving the equation, it's crucial to understand the concept of cube roots. A cube root of a number is a value that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2, because 2 * 2 * 2 = 8. Similarly, the cube root of -8 is -2, because (-2) * (-2) * (-2) = -8. Cube roots can be applied to both positive and negative numbers, which is essential to remember when solving equations like the one we're tackling today.

Step 1: Eliminating the Cube Root

The first step in solving the equation (18−x)3=−12\sqrt[3]{\left(\frac{1}{8}-x\right)}=-\frac{1}{2} is to eliminate the cube root. To do this, we need to cube both sides of the equation. Cubing a cube root effectively cancels it out, leaving us with the expression inside the cube root. This is a fundamental algebraic technique used to simplify equations involving radicals. By cubing both sides, we maintain the equality of the equation while making it easier to work with.

Let's perform this step:

((18−x)3)3=(−12)3\left(\sqrt[3]{\left(\frac{1}{8}-x\right)}\right)^3 = \left(-\frac{1}{2}\right)^3

This simplifies to:

18−x=(−12)3\frac{1}{8} - x = \left(-\frac{1}{2}\right)^3

Step 2: Calculating the Cube

Now we need to calculate the cube of −12-\frac{1}{2}. This means multiplying −12-\frac{1}{2} by itself three times: (−12)∗(−12)∗(−12)\left(-\frac{1}{2}\right) * \left(-\frac{1}{2}\right) * \left(-\frac{1}{2}\right). When multiplying fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). Remember that a negative number multiplied by a negative number results in a positive number, and a positive number multiplied by a negative number results in a negative number. This rule is crucial when dealing with exponents and negative numbers.

So, (−12)3=−18\left(-\frac{1}{2}\right)^3 = -\frac{1}{8}.

Our equation now looks like this:

18−x=−18\frac{1}{8} - x = -\frac{1}{8}

Step 3: Isolating the Variable

Our next goal is to isolate the variable x. This means getting x by itself on one side of the equation. Currently, we have 18\frac{1}{8} being subtracted from x. To isolate x, we need to eliminate the 18\frac{1}{8} term. We can do this by subtracting 18\frac{1}{8} from both sides of the equation. This is based on the principle that performing the same operation on both sides of an equation maintains the equality.

Subtracting 18\frac{1}{8} from both sides gives us:

18−x−18=−18−18\frac{1}{8} - x - \frac{1}{8} = -\frac{1}{8} - \frac{1}{8}

This simplifies to:

−x=−18−18-x = -\frac{1}{8} - \frac{1}{8}

Step 4: Combining Like Terms

Now we need to combine the like terms on the right side of the equation. We have two fractions with the same denominator, so we can simply add the numerators. Remember that when adding negative numbers, we add their absolute values and keep the negative sign. This step is essential for simplifying the equation and bringing us closer to the solution.

Combining the fractions, we get:

−x=−28-x = -\frac{2}{8}

We can simplify the fraction −28-\frac{2}{8} by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us −14-\frac{1}{4}.

So, our equation now looks like this:

−x=−14-x = -\frac{1}{4}

Step 5: Solving for x

Finally, we need to solve for x. Currently, we have -x on the left side of the equation. To get x by itself, we need to multiply both sides of the equation by -1. This will change the sign of both sides, effectively giving us the value of x. Multiplying by -1 is a common technique used to isolate variables when they have a negative coefficient.

Multiplying both sides by -1, we get:

(−1)∗(−x)=(−1)∗(−14)(-1) * (-x) = (-1) * \left(-\frac{1}{4}\right)

This simplifies to:

x=14x = \frac{1}{4}

The Solution

Therefore, the solution to the equation (18−x)3=−12\sqrt[3]{\left(\frac{1}{8}-x\right)}=-\frac{1}{2} is x=14x = \frac{1}{4}.

Verification

To ensure our solution is correct, we can substitute x=14x = \frac{1}{4} back into the original equation and see if it holds true. This is a crucial step in problem-solving, as it helps us catch any errors we might have made along the way. Verification provides confidence in our answer and ensures accuracy.

Substituting x=14x = \frac{1}{4} into the original equation, we get:

(18−14)3=−12\sqrt[3]{\left(\frac{1}{8}-\frac{1}{4}\right)}=-\frac{1}{2}

First, we need to find a common denominator for 18\frac{1}{8} and 14\frac{1}{4}, which is 8. So, we rewrite 14\frac{1}{4} as 28\frac{2}{8}.

(18−28)3=−12\sqrt[3]{\left(\frac{1}{8}-\frac{2}{8}\right)}=-\frac{1}{2}

Now we can subtract the fractions:

(−18)3=−12\sqrt[3]{\left(-\frac{1}{8}\right)}=-\frac{1}{2}

The cube root of −18-\frac{1}{8} is −12-\frac{1}{2}, because (−12)∗(−12)∗(−12)=−18\left(-\frac{1}{2}\right) * \left(-\frac{1}{2}\right) * \left(-\frac{1}{2}\right) = -\frac{1}{8}.

So, we have:

−12=−12-\frac{1}{2} = -\frac{1}{2}

Since the equation holds true, our solution x=14x = \frac{1}{4} is correct.

Conclusion

In this article, we have successfully solved the equation (18−x)3=−12\sqrt[3]{\left(\frac{1}{8}-x\right)}=-\frac{1}{2}. We began by understanding the concept of cube roots and then systematically eliminated the cube root, isolated the variable, and solved for x. We also verified our solution to ensure its accuracy. This step-by-step guide provides a clear and comprehensive approach to solving equations involving cube roots. By following these steps, you can confidently tackle similar problems in the future. Remember the key steps: eliminate the cube root, isolate the variable, and verify your solution.

Understanding these methods will build a strong foundation in mathematics. Furthermore, you'll be equipped to solve more complex problems and apply these techniques in various mathematical contexts. Keep practicing, and you'll master the art of solving equations! This detailed explanation, combined with the step-by-step approach, makes the process of solving cube root equations much more accessible and understandable.

Practice Problems

To further solidify your understanding, try solving these practice problems:

  1. (x+1)3=2\sqrt[3]{(x + 1)} = 2
  2. (2x−3)3=−1\sqrt[3]{(2x - 3)} = -1
  3. (12+x)3=12\sqrt[3]{(\frac{1}{2} + x)} = \frac{1}{2}

Working through these problems will help you reinforce the concepts and techniques discussed in this article. Remember to follow the same steps: eliminate the cube root, isolate the variable, and verify your solution. Good luck!

Additional Tips

  • Always double-check your calculations to avoid errors.
  • Pay close attention to the signs (positive and negative) of the numbers.
  • If you get stuck, try working backward from the solution to see if you can identify any mistakes.
  • Practice regularly to improve your problem-solving skills.

By incorporating these tips into your problem-solving routine, you'll become a more efficient and confident mathematician. Mathematics is a skill that improves with practice, so don't be discouraged by challenges. Embrace them as opportunities to learn and grow. The more you practice, the more comfortable and proficient you'll become in solving various types of equations. This comprehensive approach, coupled with consistent practice, will undoubtedly enhance your mathematical abilities.