Simplifying Radical Expressions A Step-by-Step Guide

In the realm of mathematics, simplifying radical expressions is a fundamental skill. It allows us to express numbers in their most concise and manageable forms. This article will guide you through the process of simplifying radical expressions, using the example expression $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ as a case study. By understanding the underlying principles and applying them systematically, you can confidently tackle similar problems.

What are Radical Expressions?

To begin, let's define what we mean by radical expressions. A radical expression is a mathematical expression that contains a radical symbol (√), which indicates the root of a number. The number under the radical symbol is called the radicand, and the small number above the radical symbol (if present) is called the index. The index tells us which root to take. For example:

• $\sqrt{9}$ is a radical expression, where 9 is the radicand and the index is 2 (since it's a square root).
• $\sqrt[3]{8}$ is a radical expression, where 8 is the radicand and the index is 3 (since it's a cube root).
• $\sqrt[4]{16}$ is a radical expression, where 16 is the radicand and the index is 4 (since it's a fourth root).

Combining Like Radicals

The key to simplifying the expression $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ lies in the concept of like radicals. Like radicals are radical expressions that have the same index and the same radicand. In other words, they are radicals that represent the same root of the same number. For example:

• $2\sqrt{3}$ and $5\sqrt{3}$ are like radicals because they both have an index of 2 (square root) and a radicand of 3.
• $3\sqrt[3]{7}$ and $-8\sqrt[3]{7}$ are like radicals because they both have an index of 3 (cube root) and a radicand of 7.
• However, $4\sqrt{5}$ and $4\sqrt[3]{5}$ are not like radicals because they have different indices (2 and 3, respectively).
• Similarly, $2\sqrt{3}$ and $2\sqrt{7}$ are not like radicals because they have different radicands (3 and 7, respectively).

Like radicals can be combined using the distributive property of multiplication over addition (or subtraction). This property states that for any numbers a, b, and c:

• a(b + c) = ab + ac
• a(b - c) = ab - ac

In the context of radical expressions, we can treat the radical part as a common factor. Therefore, we can combine like radicals by adding or subtracting their coefficients (the numbers in front of the radical symbol) while keeping the radical part unchanged. For instance:

• $2\sqrt{3} + 5\sqrt{3} = (2 + 5)\sqrt{3} = 7\sqrt{3}$
• $7\sqrt[5]{2} - 3\sqrt[5]{2} = (7 - 3)\sqrt[5]{2} = 4\sqrt[5]{2}$

Applying the Concept to the Given Expression

Now, let's apply this concept to the given expression: $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$.

First, we need to identify if the terms are like radicals. In this case, both terms have the same index (3, indicating a cube root) and the same radicand (15). Therefore, they are like radicals, and we can combine them.

To combine the like radicals, we subtract the coefficients:

$21 \sqrt[3]{15} - 9 \sqrt[3]{15} = (21 - 9) \sqrt[3]{15}$

Now, we perform the subtraction:

$(21 - 9) \sqrt[3]{15} = 12 \sqrt[3]{15}$

Thus, the simplified expression is $12 \sqrt[3]{15}$.

Step-by-step simplification

Let's break down the simplification process into clear steps:

1. Identify like radicals: Check if the terms have the same index and the same radicand. If they do, they are like radicals and can be combined.
2. Combine the coefficients: Add or subtract the coefficients of the like radicals, keeping the radical part unchanged.
3. Simplify the result: If possible, simplify the resulting expression further. This may involve simplifying the radicand or reducing the coefficients.

In our example:

1. Identify like radicals: $21 \sqrt[3]{15}$ and $9 \sqrt[3]{15}$ are like radicals because they both have an index of 3 and a radicand of 15.
2. Combine the coefficients: $21 \sqrt[3]{15} - 9 \sqrt[3]{15} = (21 - 9) \sqrt[3]{15}$
3. Simplify the result: $(21 - 9) \sqrt[3]{15} = 12 \sqrt[3]{15}$. The expression $12 \sqrt[3]{15}$ is already in its simplest form because 15 cannot be factored into perfect cubes.

Why is Simplifying Radical Expressions Important?

Simplifying radical expressions is not just a mathematical exercise; it has practical applications in various fields, including:

• Algebra: Simplified radical expressions are easier to work with when solving equations and performing other algebraic operations.
• Geometry: Radical expressions often arise when calculating lengths, areas, and volumes in geometric figures.
• Calculus: Simplifying radical expressions can make it easier to find derivatives and integrals.
• Physics and Engineering: Many physical formulas involve radical expressions, and simplifying them can make calculations more manageable.

Moreover, simplifying radical expressions promotes a deeper understanding of mathematical concepts, enhances problem-solving skills, and fosters a sense of mathematical elegance.

Common Mistakes to Avoid

When simplifying radical expressions, it's important to be aware of common mistakes that students often make. Here are a few to watch out for:

• Combining unlike radicals: As we discussed earlier, only like radicals can be combined. Avoid adding or subtracting radicals with different indices or radicands.
• Forgetting to simplify the radicand: Sometimes, the radicand can be simplified by factoring out perfect squares, cubes, or other powers. Make sure to check for this possibility.
• Incorrectly applying the distributive property: When dealing with expressions like a(b + c), remember to distribute the 'a' to both 'b' and 'c'.
• Misunderstanding the index: The index tells you which root to take. Make sure to use the correct root when simplifying.

By being mindful of these common mistakes, you can improve your accuracy and avoid errors in your calculations.

Practice Problems

To solidify your understanding of simplifying radical expressions, it's essential to practice. Here are a few problems you can try:

1. Simplify: $5 \sqrt{8} + 3 \sqrt{2}$
2. Simplify: $7 \sqrt[3]{24} - 2 \sqrt[3]{3}$
3. Simplify: $4 \sqrt{12} + 2 \sqrt{27} - \sqrt{3}$
4. Simplify: $3 \sqrt[4]{32} - \sqrt[4]{2}$

By working through these problems, you'll gain confidence in your ability to simplify radical expressions.

Conclusion: Mastering the Art of Simplifying Radicals

In this article, we have explored the process of simplifying radical expressions, focusing on the concept of like radicals and how to combine them. By understanding the underlying principles and practicing regularly, you can master this essential mathematical skill. Simplifying radical expressions is not only crucial for success in mathematics but also has practical applications in various fields. So, embrace the challenge, sharpen your skills, and enjoy the beauty of simplified expressions!

In summary, the expression $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ simplifies to $12 \sqrt[3]{15}$, which corresponds to option D.

In the vast landscape of mathematics, simplifying radical expressions stands out as a critical skill, enabling us to articulate numbers in their most succinct and manageable forms. This guide is meticulously crafted to navigate you through the intricacies of simplifying radical expressions, employing the illustrative example of $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ as our guiding beacon. By grasping the foundational principles and methodically applying them, you'll cultivate the confidence to adeptly address analogous challenges.

Unveiling Radical Expressions

Before we embark on our simplification journey, it's imperative to define the very essence of radical expressions. A radical expression, in mathematical parlance, is an expression that proudly features a radical symbol (√), emblematic of the root of a number. Nestled under this symbol lies the radicand, the number whose root we seek. The index, often a small superscript gracing the radical symbol, signifies the nature of the root—be it square, cube, or beyond. Delving into specifics:

• $\sqrt{9}$ epitomizes a radical expression, wherein 9 assumes the role of the radicand, and the invisible 2 subtly denotes the square root.
• $\sqrt[3]{8}$, another radical expression, showcases 8 as the radicand, with the index 3 explicitly indicating the cube root.
• $\sqrt[4]{16}$ continues the pattern, with 16 as the radicand and 4 directing us to the fourth root.

The Art of Combining Like Radicals

The crux of simplifying $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ lies in the elegant concept of like radicals. These are radical expressions that resonate with each other through a shared index and radicand—radicals that echo the same root of the same number. For instance:

• $2\sqrt{3}$ and $5\sqrt{3}$ harmonize as like radicals, both brandishing a square root (index of 2) and embracing the radicand 3.
• $3\sqrt[3]{7}$ and $-8\sqrt[3]{7}$ march in unison, their cube root indices (3) and radicand 7 affirming their kinship.
• Divergence appears with $4\sqrt{5}$ and $4\sqrt[3]{5}$, their indices (2 and 3, respectively) breaking the harmony.
• Similarly, $2\sqrt{3}$ and $2\sqrt{7}$ stand apart, their differing radicands (3 and 7) precluding their union.

Like radicals, akin to kindred spirits, can be melded using the distributive property, a cornerstone of algebraic manipulation. This property, a mathematical enchanter, dictates that for any trio of numbers—a, b, and c—the following holds:

• a(b + c) gracefully transforms into ab + ac.
• a(b - c) elegantly unfurls as ab - ac.

In the context of radical expressions, we perceive the radical part as a unifying thread, a common factor binding terms together. Thus, like radicals can be elegantly combined by orchestrating the coefficients—adding or subtracting them as the situation demands—while the radical part remains a steadfast constant. Behold the transformations:

• $2\sqrt{3} + 5\sqrt{3}$ metamorphoses into $(2 + 5)\sqrt{3}$, ultimately simplifying to $7\sqrt{3}$.
• $7\sqrt[5]{2} - 3\sqrt[5]{2}$ gracefully evolves into $(7 - 3)\sqrt[5]{2}$, culminating in $4\sqrt[5]{2}$.

Navigating the Given Expression

Now, let's wield this knowledge to dissect the expression at hand: $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$.

The initial step is a discerning gaze to ascertain if the terms are, indeed, like radicals. Here, both terms flaunt a cube root (index of 3) and cradle the radicand 15—a clear declaration of their like radical status.

To orchestrate their union, we perform a subtraction ballet with the coefficients:

$21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ elegantly rearranges to $(21 - 9) \sqrt[3]{15}$.

The subtraction then takes center stage:

$(21 - 9) \sqrt[3]{15}$ gracefully simplifies to $12 \sqrt[3]{15}$.

Thus, the expression, once a complex tapestry, is now a simplified serenade: $12 \sqrt[3]{15}$.

A Step-by-Step Symphony of Simplification

Let's orchestrate the simplification process into distinct movements:

1. Identifying Like Radicals: Scrutinize the terms for shared indices and radicands. If harmony exists, they are like radicals, poised for combination.
2. Harmonizing Coefficients: Conduct a symphony of addition or subtraction on the coefficients of the like radicals, while the radical part stands as an unchanging refrain.
3. Refining the Finale: Seek opportunities to further simplify the resultant expression, perhaps by untangling the radicand or paring down the coefficients.

In our specific example:

1. Like Radicals Identified: $21 \sqrt[3]{15}$ and $9 \sqrt[3]{15}$ proudly declare their like radical status, brandishing an index of 3 and cherishing the radicand 15.
2. Coefficients United: $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ gracefully transforms into $(21 - 9) \sqrt[3]{15}$.
3. The Simplified Finale: $(21 - 9) \sqrt[3]{15}$ serenely resolves to $12 \sqrt[3]{15}$. The expression, now in its minimalist form, cannot be further distilled, for 15 resists factorization into perfect cubes.

The Significance of Simplifying Radical Expressions

Simplifying radical expressions transcends mere mathematical calisthenics; it's a pragmatic skill with far-reaching applications:

• Algebra: Simplified radicals become nimble allies in equation solving and algebraic manipulations, streamlining the process.
• Geometry: Radical expressions materialize in geometric calculations, aiding in the determination of lengths, areas, and volumes.
• Calculus: The elegance of simplified radicals eases the quest for derivatives and integrals, making calculus a smoother journey.
• Physics and Engineering: Many a physical formula dons the garb of radical expressions, and their simplification renders calculations more tractable.

Moreover, the act of simplifying radicals nurtures a profound understanding of mathematical undercurrents, hones problem-solving acumen, and kindles an appreciation for mathematical grace.

Steering Clear of Common Pitfalls

The path to simplifying radicals is not without its potential pitfalls. Awareness of these common missteps is crucial:

• The Mirage of Combining Unlike Radicals: Only like radicals can unite. Resist the temptation to blend radicals with discordant indices or radicands.
• Neglecting Radicand Simplification: The radicand may harbor hidden simplicity, waiting to be unveiled through factorization into perfect powers. Overlook this, and you miss a chance for elegance.
• Distributive Property Mishaps: In expressions like a(b + c), ensure the 'a' dances its way to both 'b' and 'c'. A misstep here can lead to chaos.
• Index Misinterpretations: The index is your guide to the root. Misunderstand it, and the simplification unravels.

By vigilantly guarding against these common errors, you fortify your accuracy and ensure your calculations stand on solid ground.

Practice Makes Perfect

To truly master the art of simplifying radical expressions, practice is paramount. Here are a few exercises to hone your skills:

1. Simplify: $5 \sqrt{8} + 3 \sqrt{2}$
2. Simplify: $7 \sqrt[3]{24} - 2 \sqrt[3]{3}$
3. Simplify: $4 \sqrt{12} + 2 \sqrt{27} - \sqrt{3}$
4. Simplify: $3 \sqrt[4]{32} - \sqrt[4]{2}$

Embark on these problems, and with each solved exercise, your confidence will swell, your mastery deepen.

Epilogue: The Maestro of Radical Simplification

In this comprehensive guide, we've traversed the landscape of simplifying radical expressions, spotlighting the harmony of like radicals and their elegant combination. Armed with these principles and fueled by practice, you're poised to conquer this fundamental mathematical skill. Simplifying radicals is not just a stepping stone in mathematics; it's a gateway to broader applications. Embrace the challenge, cultivate your skills, and revel in the beauty of simplicity unveiled!

In summation, the expression $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$, when simplified, graces us with $12 \sqrt[3]{15}$, a testament to the power of mathematical elegance, perfectly captured in option D.

Within the expansive discipline of mathematics, the ability to simplify radical expressions is a cornerstone skill. It empowers us to represent numerical values in their most streamlined and comprehensible forms. This article serves as a detailed roadmap through the process of simplifying radical expressions, using the expression $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ as our central example. By grasping the fundamental concepts and implementing them methodically, you will be well-equipped to tackle analogous challenges with assurance.

Grasping the Essence of Radical Expressions

At the outset, it's crucial to define precisely what constitutes radical expressions. A radical expression is a mathematical phrase that includes a radical symbol (√), denoting the root of a number. The value under this symbol is termed the radicand, while the index, a small digit superscripted to the radical symbol (if present), indicates the specific root to be extracted. To illustrate:

• $\sqrt{9}$ is a radical expression, with 9 as the radicand and an implied index of 2, signifying a square root.
• $\sqrt[3]{8}$ represents another radical expression, where 8 is the radicand and the index 3 explicitly calls for the cube root.
• $\sqrt[4]{16}$ extends this pattern, with 16 as the radicand and 4 directing us to extract the fourth root.

The Mechanics of Combining Like Radicals

The linchpin to simplifying $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ lies in understanding the concept of like radicals. Like radicals are radical expressions sharing both the same index and the same radicand—in essence, radicals that represent an identical root of an identical number. For example:

• $2\sqrt{3}$ and $5\sqrt{3}$ qualify as like radicals, both showcasing an index of 2 (square root) and a radicand of 3.
• $3\sqrt[3]{7}$ and $-8\sqrt[3]{7}$ also align, their index of 3 (cube root) and radicand of 7 cementing their similarity.
• Contrastingly, $4\sqrt{5}$ and $4\sqrt[3]{5}$ diverge due to differing indices (2 versus 3).
• Likewise, $2\sqrt{3}$ and $2\sqrt{7}$ remain distinct, their radicands (3 and 7, respectively) precluding combination.

Like radicals can be amalgamated using the distributive property, a cornerstone of algebra. This property dictates that for any numbers a, b, and c:

• a(b + c) is equivalent to ab + ac.
• a(b - c) is equivalent to ab - ac.

In the context of radical expressions, we treat the radical component as a shared factor. Consequently, we can combine like radicals by either adding or subtracting their coefficients—the numerals preceding the radical symbol—while preserving the radical element unchanged. Examples include:

• $2\sqrt{3} + 5\sqrt{3}$ simplifies to $(2 + 5)\sqrt{3}$, ultimately yielding $7\sqrt{3}$.
• $7\sqrt[5]{2} - 3\sqrt[5]{2}$ transforms into $(7 - 3)\sqrt[5]{2}$, simplifying to $4\sqrt[5]{2}$.

Applying Principles to the Expression

Now, let's apply these principles to the expression at hand: $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$.

First, we must determine if the terms are like radicals. Both terms share an index of 3, indicating a cube root, and a radicand of 15. Thus, they are indeed like radicals and can be combined.

To combine them, we subtract the coefficients:

$21 \sqrt[3]{15} - 9 \sqrt[3]{15} = (21 - 9) \sqrt[3]{15}$

Performing the subtraction:

$(21 - 9) \sqrt[3]{15} = 12 \sqrt[3]{15}$

Thus, the simplified expression is $12 \sqrt[3]{15}$.

A Systematic Simplification Process

Let's delineate the simplification process into a series of clear steps:

1. Identify like radicals: Verify that the terms possess the same index and radicand. If so, they are like radicals and can be combined.
2. Combine coefficients: Sum or subtract the coefficients of like radicals, maintaining the radical part unchanged.
3. Simplify the result: If possible, further refine the expression. This may involve simplifying the radicand or reducing coefficients.

In our example:

1. Like radicals identified: $21 \sqrt[3]{15}$ and $9 \sqrt[3]{15}$ are like radicals, sharing an index of 3 and a radicand of 15.
2. Coefficients combined: $21 \sqrt[3]{15} - 9 \sqrt[3]{15} = (21 - 9) \sqrt[3]{15}$
3. Result simplified: $(21 - 9) \sqrt[3]{15} = 12 \sqrt[3]{15}$. The expression $12 \sqrt[3]{15}$ is in its simplest form because 15 has no factors that are perfect cubes.

The Importance of Simplifying

Simplifying radical expressions is not merely an academic exercise; it is a valuable tool across various disciplines:

• Algebra: Simplified forms are easier to manipulate in equations and other algebraic tasks.
• Geometry: Radical expressions appear in geometric calculations, such as lengths, areas, and volumes.
• Calculus: Simplification facilitates finding derivatives and integrals.
• Physics and Engineering: Many physical laws are expressed with radicals, and simplifying them eases calculations.

Furthermore, simplifying radical expressions enhances mathematical comprehension, bolsters problem-solving capabilities, and fosters mathematical elegance.

Avoiding Common Errors

Simplifying radicals can be tricky, so awareness of common errors is vital:

• Combining unlike radicals: Only like radicals can be combined. Avoid merging radicals with differing indices or radicands.
• Overlooking radicand simplification: Sometimes, the radicand can be simplified by factoring out perfect squares, cubes, or higher powers. Always check for this.
• Distributive property errors: In expressions like a(b + c), ensure 'a' is distributed to both 'b' and 'c' correctly.
• Misunderstanding the index: The index dictates which root to extract. Use the correct root for simplification.

By avoiding these pitfalls, you can improve the accuracy of your calculations.

Practice for Proficiency

To reinforce your understanding, practice is essential. Try these problems:

1. Simplify: $5 \sqrt{8} + 3 \sqrt{2}$
2. Simplify: $7 \sqrt[3]{24} - 2 \sqrt[3]{3}$
3. Simplify: $4 \sqrt{12} + 2 \sqrt{27} - \sqrt{3}$
4. Simplify: $3 \sqrt[4]{32} - \sqrt[4]{2}$

Working through these exercises will build your confidence and skill.

Conclusion: Mastering Simplification

In summary, we have explored simplifying radical expressions, focusing on like radicals and their combination. Consistent practice will lead to mastery of this essential skill. Simplifying radicals is crucial not just in mathematics but also in its real-world applications. Embrace the challenge and enjoy the elegance of simplified expressions!

In conclusion, the expression $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ simplifies to $12 \sqrt[3]{15}$, corresponding to option D.