Simplify Cube Roots Expression A Step By Step Guide

Simplifying radical expressions can sometimes seem daunting, but with a systematic approach and a good understanding of the properties of radicals, even complex-looking expressions can be tamed. In this comprehensive guide, we will walk through the step-by-step process of simplifying the expression $\sqrt[3]{320}+4 \sqrt[3]{135}$. This problem falls under the category of mathematics, specifically dealing with radicals and simplification. We'll break down each step, ensuring clarity and understanding. To begin, let's first revisit the fundamentals of simplifying radicals and the key properties that will aid us in this task. Radicals, in their essence, represent roots of numbers. The cube root, denoted by $\sqrt[3]{}$, signifies finding a number that, when multiplied by itself three times, yields the radicand (the number inside the radical). For instance, $\sqrt[3]{8} = 2$ because $2 imes 2 imes 2 = 8$. The process of simplification often involves breaking down the radicand into its prime factors and identifying perfect cubes. A perfect cube is a number that can be obtained by cubing an integer, such as 1, 8, 27, 64, and so on. The key property we'll leverage is the product property of radicals: $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. This property allows us to separate the radical of a product into the product of radicals, which is crucial for simplifying expressions where the radicand has factors that are perfect cubes. Moreover, understanding the distributive property is essential when dealing with expressions involving addition or subtraction of radicals. The distributive property states that $a(b + c) = ab + ac$, which will be useful if we need to factor out a common radical. By mastering these fundamental concepts and properties, we lay a robust foundation for tackling the simplification of $\sqrt[3]{320}+4 \sqrt[3]{135}$. The ensuing sections will meticulously detail the steps involved in simplifying each radical term and combining them to arrive at the final simplified form. This exercise not only provides a solution but also enhances our problem-solving skills in algebra and radical simplification.

Step 1: Prime Factorization of 320 and 135

To begin simplifying the expression, the first crucial step is to prime factorize the numbers inside the cube roots, namely 320 and 135. Prime factorization involves breaking down a number into its prime factors – numbers that are only divisible by 1 and themselves. This process helps us identify perfect cubes within the radicands, which we can then extract from the cube root. Let's start with 320. The prime factorization of 320 can be determined by repeatedly dividing by the smallest prime number that divides it without leaving a remainder. 320 is divisible by 2, so we have $320 = 2 imes 160$. Continuing with 160, we find $160 = 2 imes 80$. Then, $80 = 2 imes 40$, $40 = 2 imes 20$, $20 = 2 imes 10$, and $10 = 2 imes 5$. Thus, the prime factorization of 320 is $2 imes 2 imes 2 imes 2 imes 2 imes 2 imes 5$, which can be written as $2^6 imes 5$. Now, let’s move on to 135. 135 is not divisible by 2, but it is divisible by 3, so we have $135 = 3 imes 45$. Then, $45 = 3 imes 15$, and $15 = 3 imes 5$. Therefore, the prime factorization of 135 is $3 imes 3 imes 3 imes 5$, which can be written as $3^3 imes 5$. With the prime factorizations in hand, we can rewrite the original expression $\sqrt[3]{320}+4 \sqrt[3]{135}$ using these factors. So, $\sqrt[3]{320}$ becomes $\sqrt[3]{2^6 imes 5}$, and $4 \sqrt[3]{135}$ becomes $4 \sqrt[3]{3^3 imes 5}$. This step is pivotal because it sets the stage for identifying and extracting the perfect cubes from under the cube root. The next step will involve applying the properties of radicals to separate the perfect cubes and simplify the expression further. Recognizing the prime factors and their powers is crucial in simplifying radical expressions, allowing us to manipulate the terms more effectively and ultimately arrive at the simplest form. The careful breakdown of 320 and 135 into their prime factors paves the way for the subsequent simplification steps, ensuring a clear and accurate solution.

Step 2: Simplifying the Cube Roots

Having obtained the prime factorizations of 320 and 135, the next critical step is simplifying the cube roots by extracting perfect cubes. This involves applying the property $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ to separate the perfect cubes from the remaining factors under the cube root. Recall that the prime factorization of 320 is $2^6 imes 5$, and the prime factorization of 135 is $3^3 imes 5$. We can rewrite $\sqrt[3]{320}$ as $\sqrt[3]{2^6 imes 5}$. Since $2^6$ is a perfect cube (as $2^6 = (2^2)^3 = 4^3$), we can separate it from the radical: $\sqrt[3]{2^6 imes 5} = \sqrt[3]{2^6} \cdot \sqrt[3]{5}$. Now, we simplify $\sqrt[3]{2^6}$. Since $2^6 = (2^2)^3$, we have $\sqrt[3]{2^6} = 2^2 = 4$. Thus, $\sqrt[3]{320}$ simplifies to $4\sqrt[3]{5}$. Next, we simplify $4 \sqrt[3]{135}$. We rewrite it using the prime factorization of 135: $4 \sqrt[3]{3^3 imes 5}$. We can separate the perfect cube $3^3$ from the radical: $4 \sqrt[3]{3^3 imes 5} = 4 \cdot \sqrt[3]{3^3} \cdot \sqrt[3]{5}$. Since $\sqrt[3]{3^3} = 3$, we have $4 \cdot 3 \cdot \sqrt[3]{5} = 12\sqrt[3]{5}$. Now, we have simplified both terms of the original expression. $\sqrt[3]{320}$ has been simplified to $4\sqrt[3]{5}$, and $4 \sqrt[3]{135}$ has been simplified to $12\sqrt[3]{5}$. This step demonstrates the power of using prime factorization and the properties of radicals to break down complex radicals into simpler forms. By extracting the perfect cubes, we have significantly reduced the radicands, making the expression easier to manage. The next step will involve combining these simplified terms to obtain the final simplified expression. The ability to identify perfect cubes within the radicands and extract them is a fundamental skill in simplifying radicals, and this step showcases its practical application. The simplified forms $4\sqrt[3]{5}$ and $12\sqrt[3]{5}$ are now ready to be combined, leading us to the final answer.

Step 3: Combining Like Terms

After simplifying the individual cube roots, the final step is to combine like terms. In this context, like terms are terms that have the same radical part. We have simplified $\sqrt[3]{320}$ to $4\sqrt[3]{5}$ and $4\sqrt[3]{135}$ to $12\sqrt[3]{5}$. The original expression was $\sqrt[3]{320}+4 \sqrt[3]{135}$. Now we can substitute the simplified forms: $4\sqrt[3]{5} + 12\sqrt[3]{5}$. Since both terms have the same radical part, $\sqrt[3]{5}$, they are like terms and can be combined. We combine like terms by adding their coefficients. In this case, the coefficients are 4 and 12. So, we add 4 and 12: $4 + 12 = 16$. Thus, the combined term is $16\sqrt[3]{5}$. Therefore, the simplified expression for $\sqrt[3]{320}+4 \sqrt[3]{135}$ is $16\sqrt[3]{5}$. This final step showcases the importance of recognizing and combining like terms in algebraic expressions. By identifying the common radical part, we were able to consolidate the two terms into a single, simplified term. The process of combining like terms is a fundamental skill in algebra and is crucial for simplifying expressions efficiently. This step not only provides the final answer but also reinforces the principles of algebraic manipulation. In summary, we started with the expression $\sqrt[3]{320}+4 \sqrt[3]{135}$, performed prime factorization on the radicands, extracted perfect cubes to simplify the radicals, and finally, combined the like terms to arrive at the simplified form $16\sqrt[3]{5}$. The ability to navigate through these steps is a testament to understanding the properties of radicals and the principles of simplification. This comprehensive approach ensures accuracy and clarity in solving mathematical problems involving radicals. The final simplified expression, $16\sqrt[3]{5}$, represents the most concise form of the original expression, demonstrating the power of simplification techniques in mathematics.

Final Answer

In conclusion, by following a systematic approach involving prime factorization, extracting perfect cubes, and combining like terms, we have successfully simplified the given expression. The initial expression, $\sqrt[3]{320}+4 \sqrt[3]{135}$, may have seemed complex at first glance, but through careful application of mathematical principles, we have transformed it into a much simpler form. The steps undertaken included: first, breaking down the numbers under the cube roots (320 and 135) into their prime factors; second, identifying and extracting perfect cubes from the cube roots; and third, combining the resulting like terms. Each of these steps is crucial in the simplification process and highlights the importance of understanding the properties of radicals and exponents. Specifically, the ability to recognize perfect cubes and use the property $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ is instrumental in simplifying radical expressions. Furthermore, the process of combining like terms emphasizes the significance of algebraic manipulation in mathematics. By adding the coefficients of terms with the same radical part, we consolidated the expression into its most simplified form. The final result of this simplification is $16\sqrt[3]{5}$. This concise form is not only easier to work with but also provides a clear representation of the original expression's value. The simplification process not only yields the final answer but also enhances our understanding of mathematical concepts and techniques. It reinforces the importance of methodical problem-solving and the power of breaking down complex problems into manageable steps. Therefore, the simplified form of $\sqrt[3]{320}+4 \sqrt[3]{135}$ is $16\sqrt[3]{5}$. This result underscores the elegance and efficiency of mathematical simplification, allowing us to express complex quantities in their simplest and most understandable forms.