Simplifying The Cube Root Of 75a^7b^4 / 40a^13c^9

In the realm of mathematics, simplifying radical expressions is a fundamental skill. It allows us to express complex expressions in a more concise and manageable form. This article delves into the intricacies of simplifying the cube root expression $\sqrt[3]{\frac{75 a^7 b^4}{40 a^{13} c^9}}$, providing a step-by-step guide to arrive at the equivalent simplified form. We will also explore the underlying principles and techniques involved in simplifying radicals, empowering you to tackle similar problems with confidence.

Understanding the Basics of Radicals

Before we dive into the simplification process, it's crucial to grasp the fundamental concepts of radicals. A radical expression consists of a radical symbol ($\sqrt[n]{}$), a radicand (the expression under the radical), and an index (the small number 'n' indicating the root to be taken). In our case, we are dealing with a cube root, where the index is 3. Understanding these components is the cornerstone to simplify radical expressions efficiently.

Radicals are mathematical expressions that involve roots, such as square roots, cube roots, and so on. They are a way of representing the inverse operation of exponentiation. For example, the square root of a number x is a value that, when multiplied by itself, equals x. Similarly, the cube root of x is a value that, when multiplied by itself twice, equals x. To master simplifying radical expressions, it's essential to understand the properties of radicals, including how to multiply, divide, and simplify them. Understanding the properties of exponents is also crucial, as radicals and exponents are closely related. For instance, the nth root of a number can be expressed as a fractional exponent, which aids in simplifying expressions. The key to simplifying any radical expression lies in identifying perfect powers within the radicand. For cube roots, we look for factors that are perfect cubes. Simplifying radical expressions not only makes them easier to work with but also reveals the underlying mathematical structure, which is essential in various mathematical contexts, such as solving equations and understanding functions. In the context of this article, a clear understanding of radicals will help navigate the steps to simplify the given expression efficiently. By breaking down the process into manageable steps and emphasizing the core principles, anyone can learn to simplify radical expressions with confidence.

Step-by-Step Simplification of $\sqrt[3]{\frac{75 a^7 b^4}{40 a^{13} c^9}}$

Let's embark on the journey of simplifying the given expression step by step. This meticulous approach will not only lead us to the correct answer but also enhance our understanding of the simplification process. First, we focus on the numerical coefficients and then handle the variables.

1. Simplify the Fraction

Our initial step involves simplifying the fraction within the cube root. We have $\frac{75}{40}$, which can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 5. This gives us $\frac{15}{8}$.

2. Simplify the Variable Terms

Next, we turn our attention to the variable terms. We have $\frac{a^7}{a^{13}}$. Using the quotient rule of exponents ($\frac{a^m}{a^n} = a^{m-n}$), we subtract the exponents: $a^{7-13} = a^{-6}$. This term will end up in the denominator.

For the 'b' terms, we have $b^4$ in the numerator. For the 'c' terms, we have $c^9$ in the denominator.

Now, let's consolidate our simplified fraction and variable terms under the cube root:

$\sqrt[3]{\frac{15 b^4}{8 a^6 c^9}}$

3. Separate the Cube Root

We can separate the cube root of the fraction into the cube root of the numerator and the cube root of the denominator:

$\frac{\sqrt[3]{15 b^4}}{\sqrt[3]{8 a^6 c^9}}$

4. Simplify the Cube Roots

Now, we simplify each cube root individually.

• For the numerator, $\sqrt[3]{15 b^4}$, we can rewrite $b^4$ as $b^3 imes b$. The cube root of $b^3$ is simply 'b'. So, the numerator becomes $b \sqrt[3]{15 b}$.
• For the denominator, $\sqrt[3]{8 a^6 c^9}$, we find the cube roots of each factor. The cube root of 8 is 2. The cube root of $a^6$ is $a^2$ (since $(a^2)^3 = a^6$). The cube root of $c^9$ is $c^3$ (since $(c^3)^3 = c^9$). Thus, the denominator becomes $2 a^2 c^3$.

5. Combine the Simplified Terms

Putting it all together, we have:

$\frac{b \sqrt[3]{15 b}}{2 a^2 c^3}$

This is our simplified expression.

By meticulously breaking down the problem into smaller, manageable steps, we have successfully simplified the radical expression. This process highlights the importance of understanding exponent rules, simplifying fractions, and recognizing perfect cubes. Each step contributes to the final simplified form, demonstrating the beauty and precision of mathematical simplification. Furthermore, this methodical approach can be applied to a wide range of radical expressions, making it a valuable tool in your mathematical arsenal.

Matching the Simplified Expression to the Given Options

Having simplified the expression $\sqrt[3]{\frac{75 a^7 b^4}{40 a^{13} c^9}}$ to $\frac{b \sqrt[3]{15 b}}{2 a^2 c^3}$, our next crucial step is to compare this result with the options provided. This comparison will not only confirm our solution but also enhance our understanding of how different forms of the same expression can appear. The ability to recognize equivalent expressions is a fundamental skill in mathematics, particularly in algebra and calculus. It allows for flexibility in problem-solving and a deeper appreciation of mathematical structures.

Let's revisit the options:

A. $\frac{a^3 b(\sqrt[3]{15 b^2})}{2 c^3}$ B. $\frac{b(\sqrt[3]{15 b})}{2 a^2 c^3}$ C. $\frac{a^3 b(\sqrt[3]{15}}{

Upon careful examination, we can see that option B, $\frac{b(\sqrt[3]{15 b})}{2 a^2 c^3}$, perfectly matches our simplified expression. This confirms the accuracy of our step-by-step simplification process. The other options, A and C, do not match our result, highlighting the importance of meticulous simplification and comparison. This exercise reinforces the need to pay close attention to each step, from simplifying fractions and applying exponent rules to identifying and extracting perfect cubes.

Furthermore, understanding why the other options are incorrect can be just as valuable as finding the correct answer. Option A, for instance, contains an extra $a^3$ in the numerator and $b^2$ under the cube root, which are not present in our simplified form. This could indicate a misunderstanding of how to simplify variable terms or extract roots. Option C is incomplete, making it an obvious mismatch. By analyzing these discrepancies, we can identify potential areas for improvement in our understanding and technique. This process of comparison and analysis not only solidifies our current solution but also prepares us for future problems. Recognizing equivalent expressions and understanding why other options are incorrect are crucial skills for success in mathematics and related fields.

Common Mistakes to Avoid When Simplifying Radicals

Simplifying radicals can be tricky, and it's easy to make mistakes if you're not careful. Identifying and avoiding common errors is a crucial step in mastering this skill. By understanding these pitfalls, you can approach radical simplification with greater confidence and accuracy. This section highlights some frequent mistakes and provides guidance on how to steer clear of them, ensuring you arrive at the correct simplified expression.

1. Incorrectly Applying Exponent Rules

One of the most common mistakes is misapplying the rules of exponents. For example, when dividing terms with exponents, remember to subtract the exponents ($\frac{a^m}{a^n} = a^{m-n}$), not divide them. Similarly, when taking the root of a term with an exponent, divide the exponent by the index of the radical. A frequent error is to incorrectly apply the power of a power rule or to mix up the rules for multiplication and division of exponents. To avoid this, always double-check the exponent rules and practice applying them in various contexts. Understanding the underlying principles of these rules will prevent rote memorization and promote accurate application.

2. Failing to Simplify the Fraction First

Another common mistake is not simplifying the fraction inside the radical before attempting to take the root. Simplifying the fraction first makes the numbers smaller and easier to work with. This reduces the chances of making arithmetic errors and simplifies the subsequent steps. Always look for common factors in the numerator and denominator and divide them out before proceeding. For example, in our original expression, simplifying $\frac{75}{40}$ to $\frac{15}{8}$ made the rest of the simplification process smoother.

3. Not Identifying Perfect Powers

The key to simplifying radicals is identifying perfect powers within the radicand. For cube roots, this means looking for factors that are perfect cubes (e.g., 8, 27, 64). Failing to identify these perfect powers can lead to an incompletely simplified expression. Remember to factor the radicand and look for perfect powers corresponding to the index of the radical. Practice recognizing perfect squares, cubes, and higher powers to improve your speed and accuracy.

4. Incorrectly Simplifying Variables

When simplifying variables under a radical, remember to divide the exponent of the variable by the index of the radical. If the result is a whole number, the variable can be taken out of the radical. If there is a remainder, the remaining variable stays under the radical with the remainder as its exponent. For example, $\sqrt[3]{a^7}$ simplifies to $a^2 \sqrt[3]{a}$, because 7 divided by 3 is 2 with a remainder of 1. Misunderstanding this process can lead to incorrect simplification of variable terms.

5. Forgetting to Simplify the Numerical Coefficient

Sometimes, people focus so much on the variables that they forget to simplify the numerical coefficient under the radical. Always check if the numerical coefficient has any factors that are perfect powers of the index. For example, in $\sqrt[3]{16}$, the number 16 has a factor of 8, which is a perfect cube ($2^3$). Thus, $\sqrt[3]{16}$ simplifies to $2 \sqrt[3]{2}$. Overlooking this step can result in an incompletely simplified expression.

By being aware of these common mistakes and actively working to avoid them, you can significantly improve your ability to simplify radicals correctly and efficiently. Regular practice and careful attention to detail are key to mastering this skill.

Conclusion

In conclusion, simplifying radical expressions is a fundamental skill in mathematics that requires a solid understanding of exponent rules, fraction simplification, and the identification of perfect powers. By meticulously following a step-by-step approach, as demonstrated in the simplification of $\sqrt[3]{\frac{75 a^7 b^4}{40 a^{13} c^9}}$, we can arrive at the equivalent simplified form with confidence. The ability to recognize and avoid common mistakes, such as misapplying exponent rules or failing to simplify fractions, is crucial for accuracy. Moreover, comparing the simplified expression with the given options reinforces the understanding of equivalent forms and the correctness of the solution. Mastering radical simplification not only enhances problem-solving skills but also provides a deeper appreciation of mathematical structures and relationships. This skill is invaluable in various areas of mathematics and its applications, making it a worthwhile endeavor for any aspiring mathematician or scientist.