Simplifying Radical Expressions What Is The Sum Of 5(cube Root Of X) + 9(cube Root Of X)

Jul 8, 2025 by Admin 89 views

$Unlocking the Mystery of Adding Radicals: A Comprehensive Guide to Simplifying $5(\sqrt[3]{x})+9(\sqrt[3]{x})$$

Mathematics often presents us with intriguing challenges, and one such challenge arises when we encounter expressions involving radicals. Radicals, those mathematical entities denoted by the radical symbol (√), represent roots of numbers. In this comprehensive guide, we will embark on a journey to unravel the intricacies of simplifying the expression $5(\sqrt[3]{x})+9(\sqrt[3]{x})$ . This seemingly simple expression holds within it the keys to understanding the fundamental principles of radical manipulation and simplification. By the end of this exploration, you will not only be able to solve this specific problem but also possess a deeper understanding of how to approach similar mathematical challenges.

Deciphering the Expression: A Step-by-Step Approach

The expression $5(\sqrt[3]{x})+9(\sqrt[3]{x})$ presents us with a sum of two terms, each involving the cube root of x. To effectively simplify this expression, we must first recognize the common element shared by both terms: the cube root of x, denoted as ${\sqrt[3]{x}}$ . This shared element is the key to unlocking the simplification process.

Identifying the Common Radical: The Foundation of Simplification

The cornerstone of simplifying expressions involving radicals lies in identifying common radicals. In our case, both terms prominently feature ${\sqrt[3]{x}}$ . This shared radical acts as a unifying factor, allowing us to combine the terms in a meaningful way. Think of it as identifying like terms in algebraic expressions – just as we can combine 3x + 2x to get 5x, we can combine terms with the same radical.

The Distributive Property: A Powerful Tool for Simplification

With the common radical identified, we can now invoke the distributive property of multiplication over addition. This property, a fundamental principle of algebra, states that a(b + c) = ab + ac. In our context, we can view ${\sqrt[3]{x}}$ as the common factor being distributed over the coefficients 5 and 9. Applying the distributive property in reverse, we can factor out ${\sqrt[3]{x}}$ from both terms, effectively grouping the coefficients together.

Combining Like Terms: The Essence of Simplification

Having factored out the common radical, we are left with the expression (5 + 9)(\sqrt[3]{x}). The coefficients 5 and 9 are now neatly grouped within the parentheses, ready to be combined. This step mirrors the familiar process of combining like terms in algebraic expressions. Just as we would add the coefficients in 3x + 2x, we now add the coefficients 5 and 9.

The Solution Unveiled: A Step-by-Step Calculation

Let's meticulously walk through the steps of simplifying the expression, solidifying our understanding of the process:

Identify the common radical: Both terms share the cube root of x, ${\sqrt[3]{x}}$ .
Apply the distributive property: Factor out ${\sqrt[3]{x}}$ to get (5 + 9)(\sqrt[3]{x}).
Combine like terms: Add the coefficients 5 and 9 to get 14(\sqrt[3]{x}).

Therefore, the simplified expression is 14(\sqrt[3]{x}). This corresponds to none of the options provided in the original question, indicating a potential error in the answer choices.

Addressing the Answer Choices: Identifying the Discrepancy

Upon closer examination of the provided answer choices, we observe that none of them precisely match the simplified expression we derived: 14(\sqrt[3]{x}). The answer choices introduce variations involving sixth roots and exponents, which are not directly applicable to our simplified expression.

Analyzing the Incorrect Options: Understanding the Pitfalls

To gain a deeper understanding, let's analyze why the other options are incorrect:

Option A: 14( ${\sqrt[6]{x}}$ ): This option involves the sixth root of x, which is not equivalent to the cube root of x. The sixth root represents a different mathematical operation and cannot be directly substituted for the cube root.
Option B: 14( ${\sqrt[6]{x^2}}$ ): While this option includes a sixth root, it also introduces an exponent of 2 within the radical. This further deviates from our simplified expression and cannot be considered equivalent.
Option C: 14( ${\binom{\sqrt[3]{x}}{\sqrt{x}}}$ ): This option presents a binomial coefficient, which is a concept from combinatorics and is entirely unrelated to the simplification of radical expressions. This option is a clear distraction and does not hold any mathematical relevance to the problem.
Option D: 14( ${\sqrt[3]{x^2}}$ ): This option involves the cube root of x squared, which is different from the cube root of x. The exponent of 2 within the radical alters the mathematical meaning and makes this option incorrect.

The Correct Interpretation: A Mismatch in Options

Based on our step-by-step simplification, the correct answer is 14(\sqrt[3]{x}). The discrepancy between our solution and the provided answer choices suggests a potential error in the question's options. It is crucial to recognize such discrepancies and rely on sound mathematical principles to arrive at the correct answer.

Mastering Radical Simplification: Key Concepts and Techniques

Simplifying radical expressions is a fundamental skill in mathematics, with applications extending across various branches of the discipline. To master this skill, it is essential to grasp the underlying concepts and techniques.

The Essence of Radicals: Unveiling the Roots

Radicals represent the inverse operation of exponentiation. The radical symbol (√) indicates the root of a number. For instance, the square root of 9 (√9) is 3 because 3 squared (3^2) equals 9. Similarly, the cube root of 8 ( ${\sqrt[3]{8}}$ ) is 2 because 2 cubed (2^3) equals 8.

Simplifying Radicals: Extracting Perfect Powers

The key to simplifying radicals lies in identifying and extracting perfect powers from within the radical. A perfect power is a number that can be expressed as an integer raised to an integer exponent. For example, 16 is a perfect square (4^2), and 27 is a perfect cube (3^3).

Combining Radicals: The Common Radical Rule

Radicals can be combined only if they share the same index (the small number indicating the root, such as the 3 in ${\sqrt[3]{x}}$ ) and the same radicand (the expression under the radical symbol). This principle mirrors the concept of combining like terms in algebra.

Rationalizing the Denominator: Eliminating Radicals from the Bottom

In some cases, it is necessary to eliminate radicals from the denominator of a fraction. This process, known as rationalizing the denominator, involves multiplying both the numerator and denominator by a suitable expression that removes the radical from the denominator.

Expanding Our Mathematical Toolkit: Advanced Techniques and Applications

With a solid foundation in basic radical simplification, we can venture into more advanced techniques and explore the applications of radicals in diverse mathematical contexts.

Fractional Exponents: A Powerful Alternative Notation

Fractional exponents provide an alternative way to represent radicals. For instance, x^(1/2) is equivalent to √x, and x^(1/3) is equivalent to ${\sqrt[3]{x}}$ . This notation can be particularly useful when dealing with complex expressions involving radicals.

Radical Equations: Solving for the Unknown

Radical equations are equations that involve radicals. Solving these equations often requires isolating the radical term and then raising both sides of the equation to the appropriate power to eliminate the radical.

Applications in Geometry: Calculating Lengths and Distances

Radicals find widespread applications in geometry, particularly in calculating lengths and distances. The Pythagorean theorem, a cornerstone of geometry, involves square roots in its formula for determining the sides of a right triangle.

Conclusion: Embracing the Power of Simplification

In this comprehensive guide, we have meticulously explored the simplification of the expression $5(\sqrt[3]{x})+9(\sqrt[3]{x}\)$ . We have delved into the fundamental principles of radical manipulation, uncovered the step-by-step process of simplification, and addressed the discrepancy in the provided answer choices. Furthermore, we have expanded our mathematical horizons by examining advanced techniques and diverse applications of radicals.

By embracing the power of simplification, we equip ourselves with a valuable tool for navigating the intricate world of mathematics. The ability to simplify expressions not only enhances our problem-solving skills but also deepens our understanding of the underlying mathematical concepts. As we continue our mathematical journey, the principles and techniques learned here will serve as a solid foundation for tackling increasingly complex challenges.