Calculating Cube Root Of X-7 For X=-57

In the realm of mathematics, extracting roots is a fundamental operation, and among these, cube roots hold a special significance. Cube roots, denoted by the symbol $\sqrt[3]{}$, represent the value that, when multiplied by itself three times, yields the original number. This article delves into the process of finding the cube root of the expression x - 7 when x equals -57. We will explore the underlying concepts, step-by-step calculations, and practical applications of cube roots. Understanding cube roots is not only essential for mathematical proficiency but also has implications in various scientific and engineering disciplines.

Understanding Cube Roots

To understand cube roots, it's crucial to first grasp the concept of roots in general. A root of a number is a value that, when raised to a certain power, gives the original number. For instance, the square root of 9 is 3 because 3 squared (3 * 3) equals 9. Similarly, the cube root of 8 is 2 because 2 cubed (2 * 2 * 2) equals 8. The cube root operation essentially reverses the cubing operation. Mathematically, if y = x³, then the cube root of y is x, written as $\sqrt[3]{y}$ = x. Cube roots can be applied to both positive and negative numbers, as well as zero. For positive numbers, the cube root is a positive number. For negative numbers, the cube root is a negative number. The cube root of zero is zero. This property distinguishes cube roots from square roots, where the square root of a negative number is not a real number.

Cube roots have numerous applications in mathematics and various scientific fields. They are used in solving equations, simplifying expressions, and performing calculations in geometry, physics, and engineering. For example, in geometry, cube roots are used to find the side length of a cube given its volume. In physics, they appear in calculations involving volumes and densities. In engineering, cube roots are used in various design and analysis problems. Understanding cube roots provides a strong foundation for tackling more advanced mathematical concepts and real-world applications.

Step-by-Step Calculation of $\sqrt[3]{x-7}$ when x=-57

To calculate the cube root of x-7 when x=-57, we will follow a step-by-step approach. This process involves substituting the value of x into the expression, simplifying the expression, and then finding the cube root of the resulting number. This methodical approach ensures accuracy and clarity in the solution.

Step 1: Substitute the Value of x

The first step is to substitute the given value of x, which is -57, into the expression x - 7. This gives us:

-57 - 7

This is a simple arithmetic operation that forms the foundation for the rest of the calculation. Substitution is a fundamental technique in algebra, allowing us to evaluate expressions and solve equations by replacing variables with specific values.

Step 2: Simplify the Expression

Next, we simplify the expression -57 - 7. This involves subtracting 7 from -57, which results in:

-57 - 7 = -64

So, the expression x - 7 when x is -57 simplifies to -64. This step is crucial because it reduces the original expression to a single number, making it easier to find the cube root. Accurate arithmetic is essential in this step to avoid errors in the final result.

Step 3: Find the Cube Root

Now, we need to find the cube root of -64. The cube root of a number is the value that, when multiplied by itself three times, equals the original number. In this case, we are looking for a number that, when cubed, gives -64. We can express this as:

$\sqrt[3]{-64}$

Since -64 is a negative number, we know that the cube root will also be negative. We need to find a number that, when multiplied by itself three times, results in 64. We know that:

4 * 4 * 4 = 64

Therefore, the cube root of 64 is 4. Since we are looking for the cube root of -64, the answer will be -4. We can verify this by cubing -4:

(-4) * (-4) * (-4) = -64

Thus, the cube root of -64 is -4. This completes the calculation, giving us the final answer.

Detailed Explanation of the Solution

In solving this problem, we followed a clear and logical progression. We started by substituting the value of x into the expression, which is a fundamental step in algebra. Then, we simplified the expression using basic arithmetic operations. Finally, we found the cube root of the simplified number. This process highlights the importance of breaking down complex problems into smaller, manageable steps.

The cube root of -64 is -4 because (-4) * (-4) * (-4) equals -64. This demonstrates the definition of a cube root: the number that, when multiplied by itself three times, yields the original number. Understanding this concept is crucial for working with cube roots and other types of roots.

It's important to note that cube roots can be found for both positive and negative numbers. This is different from square roots, where the square root of a negative number is not a real number. The cube root of a negative number is a negative number, as seen in this example. This property makes cube roots useful in various applications where negative values are encountered.

The step-by-step approach used in this calculation can be applied to other similar problems. By breaking down the problem into smaller steps, we can ensure accuracy and clarity in the solution. This method is particularly useful when dealing with more complex expressions or equations.

Practical Applications of Cube Roots

Cube roots are not just theoretical mathematical concepts; they have numerous practical applications in various fields. Understanding these applications can help to appreciate the importance of cube roots in real-world scenarios.

Geometry

In geometry, cube roots are used to find the side length of a cube given its volume. The volume V of a cube with side length s is given by the formula:

V = s³

To find the side length s, we take the cube root of the volume:

s = $\sqrt[3]{V}$

For example, if a cube has a volume of 216 cubic centimeters, the side length is the cube root of 216, which is 6 centimeters. This application is essential in various fields, including architecture, engineering, and construction, where calculating volumes and dimensions of three-dimensional objects is crucial.

Physics

In physics, cube roots appear in calculations involving volumes and densities. For instance, the density ρ of an object is defined as its mass m divided by its volume V:

ρ = m / V

If we know the mass and density of an object and want to find its volume, we can rearrange the formula to:

V = m / ρ

If the object is a sphere, its volume is given by:

V = (4/3)πr³

where r is the radius of the sphere. To find the radius, we need to take the cube root of a term involving the volume. This application is vital in material science, fluid dynamics, and other areas of physics where volumetric calculations are necessary.

Engineering

In engineering, cube roots are used in various design and analysis problems. For example, in mechanical engineering, the stress and strain in materials under load can involve cube root calculations. In chemical engineering, cube roots might be used in calculations involving the volumes of reactors or the sizes of particles in a suspension. In civil engineering, cube roots can appear in calculations related to the stability of structures or the flow of fluids in pipes. These applications highlight the importance of cube roots in solving practical engineering problems and designing efficient systems.

Conclusion

In conclusion, finding the cube root of x - 7 when x = -57 involves a straightforward process of substitution, simplification, and root extraction. By substituting -57 for x, we simplified the expression to -64. Then, we determined that the cube root of -64 is -4. This calculation demonstrates the fundamental principles of cube roots and their application in algebra.

Cube roots are a crucial concept in mathematics, with applications spanning various fields such as geometry, physics, and engineering. Understanding cube roots allows us to solve problems involving volumes, densities, and other three-dimensional quantities. The step-by-step approach outlined in this article provides a clear method for calculating cube roots, ensuring accuracy and clarity in problem-solving.

Whether you are a student learning algebra or a professional working in a technical field, a solid understanding of cube roots is essential. This article has provided a comprehensive guide to finding cube roots, complete with examples and practical applications. By mastering these concepts, you can enhance your mathematical skills and tackle a wide range of real-world problems.