Simplifying Radicals How To Simplify $\sqrt[3]{72 T^{18} W^{14}}$

This article will guide you through simplifying the expression $\sqrt[3]{72 t^{18} w^{14}}$ into its simplest radical form. We will break down each component of the expression, apply relevant mathematical rules, and arrive at the simplified result. This process involves understanding how to handle cube roots, exponents, and variable terms within a radical expression. Simplifying radical expressions is a fundamental skill in algebra, and mastering it will significantly enhance your problem-solving abilities in more advanced mathematical contexts. Let’s dive into each step to ensure a clear and thorough understanding.

Understanding the Basics of Radical Simplification

To effectively simplify $\sqrt[3]{72 t^{18} w^{14}}$, we need to understand the fundamental principles of radical simplification. A radical expression consists of a radical symbol ($\sqrt[n]{}$), a radicand (the expression under the radical), and an index (n, indicating the root to be taken). In our case, the index is 3, signifying a cube root, and the radicand is $72 t^{18} w^{14}$. The primary goal of simplifying radicals is to remove any perfect nth powers from the radicand. This involves factoring the radicand and extracting any factors that can be written as perfect cubes. Additionally, we aim to ensure that the radicand contains no fractions and that the index is as small as possible. Simplifying radicals not only makes expressions easier to work with but also reveals the underlying structure of mathematical expressions. Each term inside the radical must be examined carefully to identify factors that can be extracted. This process often requires a combination of numerical factorization and application of exponent rules. The simplified form allows for easier comparison and combination of like terms, which is crucial in various algebraic manipulations.

Breaking Down the Radicand: Numerical and Variable Components

In our expression $\sqrt[3]{72 t^{18} w^{14}}$, the radicand is composed of both numerical and variable parts. The numerical part is 72, and the variable parts are $t^{18}$ and $w^{14}$. To simplify the cube root, we must address each component separately. First, we factorize 72 to identify any perfect cube factors. The prime factorization of 72 is $2^3 \cdot 3^2$ or $8 \cdot 9$. Here, 8 ($2^3$) is a perfect cube, which can be extracted from the cube root. Next, we deal with the variable components. For $t^{18}$, we can directly apply the exponent rule for radicals. Since 18 is divisible by 3, $t^{18}$ is a perfect cube. Specifically, $\sqrt[3]{t^{18}} = t^{18/3} = t^6$. However, for $w^{14}$, the exponent 14 is not divisible by 3. We need to rewrite $w^{14}$ as a product of a perfect cube and a remaining power. We can express $w^{14}$ as $w^{12} \cdot w^2$, where $w^{12}$ is a perfect cube because 12 is divisible by 3. Thus, $\sqrt[3]{w^{12}} = w^{12/3} = w^4$. The remaining $w^2$ will stay inside the cube root. This breakdown allows us to systematically extract perfect cubes from the radicand, making the simplification process more manageable and accurate. Understanding the interplay between numerical factorization and exponent manipulation is key to mastering radical simplification.

Step-by-Step Simplification of $\sqrt[3]{72 t^{18} w^{14}}$

Now, let’s walk through the step-by-step simplification of the expression $\sqrt[3]{72 t^{18} w^{14}}$.

1. Factor the numerical part: As discussed, the prime factorization of 72 is $2^3 \cdot 3^2$. We rewrite 72 as $8 \cdot 9$, where 8 is the perfect cube $2^3$.
2. Rewrite the expression: Substitute the factorization of 72 into the original expression: $\sqrt[3]{72 t^{18} w^{14}} = \sqrt[3]{2^3 \cdot 3^2 \cdot t^{18} \cdot w^{14}}$.
3. Break down the variable terms: We've already determined that $t^{18}$ is a perfect cube. For $w^{14}$, we rewrite it as $w^{12} \cdot w^2$.
4. Rewrite with broken down variable terms: Now the expression looks like this: $\sqrt[3]{2^3 \cdot 3^2 \cdot t^{18} \cdot w^{12} \cdot w^2}$.
5. Apply the cube root: Extract the cube roots of the perfect cubes. $\sqrt[3]{2^3} = 2$, $\sqrt[3]{t^{18}} = t^6$, and $\sqrt[3]{w^{12}} = w^4$. The remaining terms $3^2$ and $w^2$ will stay inside the cube root.
6. Write the simplified expression: Combine the extracted terms and the remaining terms under the cube root: $2 t^6 w^4 \sqrt[3]{3^2 w^2}$.
7. Final simplification: Simplify $3^2$ to 9. The final simplified expression is $2 t^6 w^4 \sqrt[3]{9 w^2}$.

Each step is crucial in ensuring that the expression is correctly simplified. Paying close attention to both numerical and variable components allows for a systematic and accurate simplification process. The final result represents the most simplified form of the original expression, with all possible perfect cubes extracted from the radicand.

Detailed Explanation of Each Simplification Step

To further clarify the simplification process, let's delve into a detailed explanation of each step. Starting with factoring the numerical part, we identified that 72 could be broken down into $2^3 \cdot 3^2$. Recognizing $2^3$ (which is 8) as a perfect cube is essential because it allows us to extract 2 from the cube root. The remaining factor, $3^2$ (which is 9), cannot be further simplified within a cube root context, so it will remain inside the radical. When we move to the variable terms, $t^{18}$ is straightforward since 18 is divisible by 3. The cube root of $t^{18}$ is $t^{18/3} = t^6$. This direct application of the exponent rule simplifies the variable term efficiently. The term $w^{14}$ requires a bit more attention. Since 14 is not divisible by 3, we rewrite $w^{14}$ as $w^{12} \cdot w^2$. Here, $w^{12}$ is a perfect cube, and its cube root is $w^{12/3} = w^4$. The remaining $w^2$ stays under the radical because it does not form a perfect cube. After breaking down both numerical and variable parts, we combine the extracted cube roots: 2 from $2^3$, $t^6$ from $t^{18}$, and $w^4$ from $w^{12}$. These terms are placed outside the radical. Inside the radical, we are left with $3^2 \cdot w^2$, which simplifies to $9w^2$. Thus, the final simplified expression is $2t^6w^4\sqrt[3]{9w^2}$. Understanding the rationale behind each step ensures that you can apply these techniques to a variety of radical simplification problems.

Final Simplified Form: $2 t^6 w^4 \sqrt[3]{9 w^2}$

After meticulously breaking down and simplifying the expression $\sqrt[3]{72 t^{18} w^{14}}$, we arrive at the final simplified form: $2 t^6 w^4 \sqrt[3]{9 w^2}$. This result represents the original expression with all possible perfect cube factors extracted from the radicand. The coefficient 72 has been reduced to its simplest form within the cube root context, the variable $t$ has been fully simplified due to its exponent being a multiple of 3, and the variable $w$ has been partially simplified, with the remaining non-cube factor staying inside the radical. This simplified form is not only mathematically equivalent to the original expression but also easier to interpret and use in further calculations. The process of simplification has made the expression more manageable and clear. The key takeaway from this exercise is the systematic approach to radical simplification: identifying perfect cube factors, extracting them from the radical, and leaving behind the irreducible components. Mastering this technique is crucial for success in algebra and beyond, as simplified expressions often reveal underlying mathematical structures and facilitate problem-solving in various contexts.

In conclusion, by following the principles of radical simplification, we have successfully transformed $\sqrt[3]{72 t^{18} w^{14}}$ into its simplest radical form, which is $2 t^6 w^4 \sqrt[3]{9 w^2}$. This process underscores the importance of understanding factorization, exponent rules, and the properties of radicals. The ability to simplify such expressions is a valuable skill in mathematics, providing clarity and efficiency in more complex calculations.