Simplifying Cube Root: (-81k^3)/(j^3) Solution

Let's dive into simplifying the cube root of the expression (-81k^3)/(j3). This is a classic algebra problem that combines the properties of exponents and roots. We'll break it down step by step to make it easy to understand. This detailed guide will not only provide the solution but also explain the underlying principles, ensuring you grasp the concepts thoroughly.

Understanding Cube Roots

Before we tackle the main problem, it's crucial to understand what a cube root is. A cube root of a number is a value that, when multiplied by itself three times, gives you the original number. For example, the cube root of 8 is 2 because 2 * 2 * 2 = 8. Mathematically, we represent the cube root of a number x as ∛x. When dealing with variables and negative numbers, understanding the rules becomes even more critical.

The cube root operation is the inverse of cubing a number. This means that if you take the cube root of a number that has been cubed, you will return to the original number. For instance, ∛(a^3) = a. This property is fundamental when simplifying expressions involving radicals and exponents. In the case of negative numbers, the cube root of a negative number is negative because a negative number multiplied by itself three times results in a negative number. For example, ∛(-8) = -2 because -2 * -2 * -2 = -8. This is a crucial distinction from square roots, where the square root of a negative number is not a real number.

When simplifying expressions, it's also important to remember the properties of radicals. One key property is that the cube root of a product is the product of the cube roots. Mathematically, this is expressed as ∛(ab) = ∛a * ∛b. Similarly, the cube root of a quotient is the quotient of the cube roots, expressed as ∛(a/b) = ∛a / ∛b. These properties allow us to break down complex expressions into simpler parts, making them easier to simplify. Understanding these fundamental principles of cube roots is essential for solving problems involving radicals and exponents, especially when dealing with expressions that include variables and negative numbers. With a solid grasp of these concepts, you'll be well-equipped to tackle more complex problems in algebra and beyond.

Breaking Down the Expression

The expression we need to simplify is ∛((-81k^3)/(j3)). To make this easier, we can break it down using the properties of radicals. The cube root of a fraction can be expressed as the cube root of the numerator divided by the cube root of the denominator. So, we can rewrite the expression as:

∛((-81k^3)/(j3)) = ∛(-81k^3) / ∛(j^3)

This separation allows us to focus on the numerator and the denominator individually. Now, let's further break down the numerator, ∛(-81k^3). We can express -81 as a product of its factors, where one of the factors is a perfect cube. The number -81 can be written as -27 * 3, where -27 is a perfect cube (-3 * -3 * -3 = -27). So, we can rewrite the numerator as:

∛(-81k^3) = ∛(-27 * 3 * k^3)

Using the property that the cube root of a product is the product of the cube roots, we can further separate this into:

∛(-27 * 3 * k^3) = ∛(-27) * ∛(3) * ∛(k^3)

Now, we can evaluate the cube roots of the perfect cubes. We know that ∛(-27) = -3 and ∛(k^3) = k. So, the numerator simplifies to:

-3 * ∛(3) * k = -3k∛(3)

Next, let's consider the denominator, ∛(j^3). This is a straightforward cube root, and we know that ∛(j^3) = j. So, the denominator simplifies to j. By breaking down the expression in this way, we make it easier to handle each part separately and apply the properties of radicals and exponents effectively. This methodical approach is key to simplifying complex expressions in mathematics. Understanding each step ensures that you not only arrive at the correct answer but also grasp the underlying concepts and techniques involved.

Simplifying the Numerator

Let's focus on simplifying the numerator, ∛(-81k^3). Our goal is to break down the expression into its simplest form by identifying perfect cubes within the radical. The first step is to recognize that -81 can be factored into -27 multiplied by 3. Here, -27 is a perfect cube because -3 * -3 * -3 = -27. We can rewrite the expression as:

∛(-81k^3) = ∛(-27 * 3 * k^3)

Now, we apply the property that the cube root of a product is the product of the cube roots. This allows us to separate the expression into individual cube roots:

∛(-27 * 3 * k^3) = ∛(-27) * ∛(3) * ∛(k^3)

Next, we evaluate the cube roots of the perfect cubes. We know that ∛(-27) is -3 because -3 cubed is -27. Similarly, ∛(k^3) is simply k because k cubed is k^3. So, we have:

∛(-27) * ∛(3) * ∛(k^3) = -3 * ∛(3) * k

Rewriting this in a more conventional format, we get:

-3k∛(3)

This is the simplified form of the numerator. By breaking down the original expression into its constituent factors and applying the properties of cube roots, we have successfully reduced it to a simpler form. The key to this process is identifying perfect cubes within the radical and separating the expression into manageable parts. This approach is widely used in simplifying radical expressions and is an essential skill in algebra. Understanding each step of this process ensures a solid grasp of the underlying mathematical principles, enabling you to tackle more complex problems with confidence. With the numerator simplified, we can now move on to the denominator and complete the simplification of the entire expression.

Simplifying the Denominator

Now, let's tackle the denominator of the original expression, which is ∛(j^3). Simplifying this is much more straightforward than the numerator because we are dealing with a perfect cube. The cube root of a variable raised to the power of 3 is simply the variable itself. In mathematical terms:

∛(j^3) = j

This is a fundamental property of cube roots and exponents. When you take the cube root of a quantity that is cubed, the cube root operation effectively undoes the cubing operation, leaving you with the original quantity. There's no need for further factorization or breaking down the expression; it's already in its simplest form.

The simplicity of this step highlights the importance of recognizing perfect cubes when simplifying radical expressions. In this case, j^3 is a perfect cube, making its cube root a straightforward simplification. This step is a crucial part of the overall simplification process, as it reduces the complexity of the denominator and allows us to combine it with the simplified numerator to get the final result. Understanding this basic principle is essential for dealing with radicals and exponents effectively. By mastering these fundamental concepts, you can confidently simplify more complex expressions and solve a wide range of mathematical problems. With the denominator now simplified, we can combine it with the simplified numerator to arrive at the final answer.

Combining Simplified Parts

With both the numerator and the denominator simplified, we can now combine them to find the simplified form of the original expression. Recall that we broke down the expression as follows:

∛((-81k^3)/(j3)) = ∛(-81k^3) / ∛(j^3)

We simplified the numerator, ∛(-81k^3), to -3k∛(3), and the denominator, ∛(j^3), to j. Now, we substitute these simplified forms back into the expression:

∛((-81k^3)/(j3)) = (-3k∛(3)) / j

This is the simplified form of the original expression. We have successfully reduced a complex cube root expression into a more manageable form by breaking it down into its constituent parts, simplifying each part separately, and then combining them. The key steps in this process were identifying perfect cubes, applying the properties of radicals, and simplifying the numerator and denominator individually. This methodical approach is a fundamental technique in algebra and is crucial for simplifying a wide range of expressions involving radicals and exponents.

By understanding each step of this process, you can confidently tackle similar problems and gain a deeper understanding of algebraic manipulation. The ability to simplify complex expressions is an essential skill in mathematics, and this example provides a clear illustration of how to approach such problems systematically. The final simplified expression, (-3k∛(3)) / j, represents the cube root of the original expression in its simplest form, demonstrating the power of breaking down problems into manageable parts and applying fundamental mathematical principles.

Final Solution

Therefore, the simplified form of the expression ∛((-81k^3)/(j3)) is:

(-3k∛(3)) / j

This is our final answer. We arrived at this solution by systematically breaking down the expression, simplifying the numerator and denominator separately, and then combining the results. This process involved understanding and applying the properties of cube roots, identifying perfect cubes, and simplifying radical expressions. Each step was crucial in reducing the complexity of the original expression and arriving at the final simplified form.

This solution demonstrates the importance of a methodical approach to simplifying mathematical expressions. By breaking down complex problems into smaller, more manageable parts, we can apply the appropriate mathematical principles and techniques to each part individually. This not only makes the problem easier to solve but also helps to ensure accuracy in the final result. The ability to simplify expressions like this is a fundamental skill in algebra and is essential for further studies in mathematics and related fields.

In summary, the key takeaways from this problem are the understanding of cube roots, the ability to identify and extract perfect cubes from radicals, and the application of the properties of radicals to simplify expressions. With a solid grasp of these concepts, you can confidently tackle a wide range of problems involving radicals and exponents. The final solution, (-3k∛(3)) / j, represents the culmination of these efforts and demonstrates the power of systematic problem-solving in mathematics.

In conclusion, simplifying cube root expressions involves a methodical approach, including breaking down the expression, identifying perfect cubes, and applying radical properties. By understanding each step, you can confidently solve similar problems. For further exploration of radical expressions and simplification techniques, visit Khan Academy's Algebra Resources.