Simplifying Radicals A Step-by-Step Guide To √10ab × √15a²b³

Introduction

In this comprehensive guide, we delve into the simplification of radical expressions, specifically focusing on the expression ${\sqrt{10ab} \times \sqrt{15a^2b^3}}$. Mastering the art of simplifying radicals is crucial in mathematics, as it allows for a more concise and manageable form of expressions, which can be invaluable in various mathematical contexts. This article will meticulously walk you through the step-by-step process, ensuring you grasp the underlying principles and techniques involved. We'll start by understanding the basic properties of radicals, then proceed to break down the given expression, and finally, simplify it to its simplest radical form. This journey will equip you with the skills to tackle similar problems with confidence and precision.

Understanding Radicals and Their Properties

Before we dive into the simplification process, it's essential to have a firm grasp of what radicals are and the fundamental properties that govern their behavior. A radical, in its simplest form, is a root of a number. The most common radical is the square root, denoted by the symbol ${\sqrt{}}$, which represents the non-negative number that, when multiplied by itself, equals the number under the radical sign (the radicand). For instance, ${\sqrt{9} = 3}$ because 3 multiplied by itself equals 9. Radicals also include cube roots, fourth roots, and so on, each denoted by an index number indicating the degree of the root. Understanding these basics is the cornerstone of simplifying radical expressions effectively.

One of the most crucial properties we'll leverage in simplifying our expression is the product property of radicals. This property states that the square root of a product is equal to the product of the square roots of the individual factors. Mathematically, this can be expressed as ${\sqrt{ab} = \sqrt{a} \times \sqrt{b}}$, where a and b are non-negative numbers. This property allows us to break down complex radicals into simpler components, making the simplification process more manageable. Another essential concept is identifying perfect squares within the radicand. A perfect square is a number that can be expressed as the square of an integer (e.g., 4, 9, 16, 25). Recognizing perfect squares enables us to extract their square roots, further simplifying the radical expression. For example, in ${\sqrt{36}}$, 36 is a perfect square (6 squared), so ${\sqrt{36}}$ simplifies to 6.

Furthermore, when dealing with variables inside radicals, we apply similar principles. For a variable raised to an even power, its square root is simply the variable raised to half that power. For instance, ${\sqrt{x^4} = x^2}$ because ${(x^2)^2 = x^4}$. If the variable is raised to an odd power, we can separate out the highest even power and a single factor of the variable. For example, ${\sqrt{x^5}}$ can be rewritten as ${\sqrt{x^4 \times x}}$, which simplifies to ${x^2\sqrt{x}}$. These foundational principles form the bedrock of simplifying radicals, enabling us to methodically approach and solve complex expressions. With a clear understanding of these concepts, we are well-prepared to tackle the given problem and simplify ${\sqrt{10ab} \times \sqrt{15a^2b^3}}$ to its simplest radical form.

Step-by-Step Simplification of ${\sqrt{10ab} \times \sqrt{15a^2b^3}}$

Initial Multiplication

Let's embark on the journey of simplifying the radical expression ${\sqrt{10ab} \times \sqrt{15a^2b^3}}$. The first step in this process involves applying the product property of radicals, which we discussed earlier. This property allows us to combine the two separate radicals into a single radical by multiplying the radicands (the expressions inside the square root). By doing so, we consolidate the expression and pave the way for further simplification. This initial multiplication is a crucial step in unraveling the complexity of the expression and bringing it closer to its simplest form.

Applying the product property, we get:

${\sqrt{10ab} \times \sqrt{15a^2b^3} = \sqrt{(10ab) \times (15a^2b^3)}}$

Now, we multiply the terms inside the radical:

${\sqrt{(10ab) \times (15a^2b^3)} = \sqrt{10 \times 15 \times a \times a^2 \times b \times b^3}}$

Performing the multiplication, we have:

${\sqrt{150a^3b^4}}$

This resulting radical, ${\sqrt{150a^3b^4}}$, now encapsulates the product of the original two radicals. It is a single, consolidated radical expression that we can further simplify. The next phase of our journey involves breaking down this radicand into its constituent factors, particularly looking for perfect squares. Identifying and extracting perfect squares is a key strategy in simplifying radicals, as it allows us to reduce the expression to its most basic components. By meticulously factoring the radicand, we aim to isolate perfect square factors, which can then be easily simplified. This process transforms the radical into a more manageable form, paving the way for the final simplification steps.

Factoring the Radicand

Having obtained ${\sqrt{150a^3b^4}}$, the next crucial step is to factor the radicand, which is the expression inside the square root (150a³b⁴). This process involves breaking down the number and variables into their prime factors or perfect square components. Factoring the radicand is a strategic maneuver that allows us to identify and extract any perfect squares, which are essential for simplifying the radical expression. By meticulously factoring, we aim to distill the radicand into its most basic parts, making the subsequent simplification steps more straightforward.

Let's begin by factoring the numerical coefficient, 150. We look for the largest perfect square that divides 150. The prime factorization of 150 is 2 × 3 × 5². We can see that 25 (5²) is a perfect square factor of 150. Therefore, we can rewrite 150 as 25 × 6. Now, let's turn our attention to the variable terms. For a³, we can rewrite it as a² × a, where a² is a perfect square. For b⁴, it is already a perfect square since it can be written as (b²)². Applying these factorizations, we rewrite the radicand as follows:

${150a^3b^4 = 25 \times 6 \times a^2 \times a \times b^4}$

Now, substituting this back into our radical expression, we have:

${\sqrt{150a^3b^4} = \sqrt{25 \times 6 \times a^2 \times a \times b^4}}$

By rewriting the radicand in this manner, we have successfully isolated the perfect square factors (25, a², and b⁴). This strategic factorization sets the stage for the next phase of simplification, where we will apply the product property of radicals in reverse to separate these perfect squares and simplify them individually. This step is pivotal in reducing the radical expression to its simplest form, as it allows us to extract the square roots of the perfect square factors, thereby simplifying the overall expression. The systematic factoring of the radicand is a key technique in radical simplification, enabling us to transform complex expressions into more manageable and understandable forms.

Extracting Perfect Squares

With the radicand factored into ${25 \times 6 \times a^2 \times a \times b^4}$, the next crucial step in simplifying ${\sqrt{150a^3b^4}}$ is to extract the perfect squares. As we identified in the previous step, 25, a², and b⁴ are the perfect square factors within our radicand. Extracting these perfect squares involves applying the product property of radicals in reverse, effectively separating the square roots of the perfect squares from the remaining factors. This process is a pivotal moment in the simplification, as it allows us to reduce the radical expression to its most basic components.

Applying the product property in reverse, we can rewrite the radical as a product of individual radicals:

${\sqrt{25 \times 6 \times a^2 \times a \times b^4} = \sqrt{25} \times \sqrt{6} \times \sqrt{a^2} \times \sqrt{a} \times \sqrt{b^4}}$

Now, we simplify each of the square roots of the perfect squares:

• ${\sqrt{25} = 5}$ since 5 × 5 = 25
• ${\sqrt{a^2} = a}$ because a × a = a²
• ${\sqrt{b^4} = b^2}$ since b² × b² = b⁴

The remaining factors, 6 and a, do not have perfect square factors and will remain inside the radical. Substituting the simplified square roots back into the expression, we get:

${5 \times \sqrt{6} \times a \times \sqrt{a} \times b^2}$

Now, we rearrange the terms to group the simplified components together:

${5 \times a \times b^2 \times \sqrt{6} \times \sqrt{a}}$

Finally, we combine the remaining radicals using the product property:

${5ab^2\sqrt{6a}}$

Thus, by systematically extracting the perfect squares, we have successfully simplified the radical expression. The result, ${5ab^2\sqrt{6a}}$, is now in its simplest radical form, as there are no remaining perfect square factors under the radical sign. This step-by-step extraction of perfect squares is a cornerstone technique in simplifying radicals, allowing us to transform complex expressions into their most concise and understandable forms. With the expression now in its simplest form, we have completed the simplification process.

Final Simplified Form

After meticulously working through the steps of multiplication, factoring, and extracting perfect squares, we have arrived at the final simplified form of the expression ${\sqrt{10ab} \times \sqrt{15a^2b^3}}$. The journey from the initial complex expression to its simplified form has been a testament to the power of understanding and applying the properties of radicals. By breaking down the problem into manageable steps, we have successfully navigated the intricacies of radical simplification.

The final simplified form of the expression is:

${5ab^2\sqrt{6a}}$

This result, ${5ab^2\sqrt{6a}}$, represents the most concise and manageable form of the original expression. It is a culmination of our efforts in identifying and extracting perfect square factors, leaving only irreducible components under the radical sign. This simplified form is not only aesthetically pleasing but also practically advantageous, as it makes the expression easier to work with in various mathematical contexts. For instance, if we were to substitute specific values for a and b, the simplified form would allow for a much more straightforward calculation compared to the original expression.

In this simplified form, the coefficient outside the radical is 5ab², which is a product of integers and variables raised to integer powers. The radicand, 6a, contains no perfect square factors other than 1, ensuring that the radical is in its simplest form. The variable b is squared outside the radical, indicating that we have successfully extracted the maximum possible power of b from the original expression. The presence of a under the radical signifies that we were unable to extract a perfect square factor for a, as it appeared to an odd power in the original radicand. This meticulous simplification process ensures that we have captured all the essential elements of the original expression in its most reduced form.

Conclusion

In conclusion, we have successfully simplified the radical expression ${\sqrt{10ab} \times \sqrt{15a^2b^3}}$ to its simplest form, which is ${5ab^2\sqrt{6a}}$. This journey has highlighted the importance of understanding and applying the fundamental properties of radicals, including the product property and the identification of perfect squares. By breaking down the problem into manageable steps, we have demonstrated a systematic approach to simplifying radicals, which can be applied to a wide range of similar expressions.

We began by combining the two radicals into a single radical using the product property, resulting in ${\sqrt{150a^3b^4}}$. We then meticulously factored the radicand, identifying the perfect square factors within 150, a³, and b⁴. This factorization allowed us to rewrite the expression as ${\sqrt{25 \times 6 \times a^2 \times a \times b^4}}$. Next, we extracted the square roots of the perfect square factors, simplifying ${\sqrt{25}}$ to 5, ${\sqrt{a^2}}$ to a, and ${\sqrt{b^4}}$ to b². This process yielded the simplified expression ${5ab^2\sqrt{6a}}$, which is the final answer.

This exercise underscores the significance of mastering radical simplification techniques. Simplified radical forms are not only more aesthetically pleasing but also more practical for various mathematical operations, such as evaluating expressions, solving equations, and performing further algebraic manipulations. The ability to simplify radicals efficiently is a valuable skill in mathematics, enabling us to work with expressions in their most manageable forms.

The process we have demonstrated here is a general approach that can be applied to simplify a wide variety of radical expressions. By focusing on the key steps of combining radicals, factoring radicands, identifying perfect squares, and extracting their square roots, you can confidently tackle radical simplification problems. Remember to always look for the largest perfect square factors and to simplify each component systematically. With practice and a solid understanding of the underlying principles, you can master the art of simplifying radicals and excel in your mathematical endeavors. This detailed walkthrough serves as a valuable resource for anyone looking to enhance their understanding and skills in simplifying radical expressions.