Simplifying The Expression $\sqrt{(2)}^2+\sqrt[3]{9 \times 7}-\sqrt[3]{(2)^6}$ A Step-by-Step Guide

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Evaluate the expression: $\sqrt{(2)}^2+\sqrt[3]{9 \times 7}-\sqrt[3]{(2)^6}$

Introduction

In the realm of mathematics, expressions often appear complex at first glance, but with careful application of mathematical principles and simplification techniques, they can be elegantly resolved. This article delves into a specific mathematical expression: (2)2+9×73−(2)63\sqrt{(2)}^2+\sqrt[3]{9 \times 7}-\sqrt[3]{(2)^6}. Our goal is to break down each component of the expression, apply relevant mathematical rules, and arrive at a simplified, final answer. We will explore the concepts of square roots, cube roots, exponents, and basic arithmetic operations, providing a step-by-step solution that is easy to follow and understand. This exploration is not just about finding the answer; it's about understanding the process and the underlying mathematical concepts that govern the solution. Let's embark on this mathematical journey and unravel the intricacies of this expression.

Breaking Down the Expression

The expression we are tackling is (2)2+9×73−(2)63\sqrt{(2)}^2+\sqrt[3]{9 \times 7}-\sqrt[3]{(2)^6}. To solve this, we'll address each term individually before combining the results. The expression consists of three main terms: the first involving a square root and a square, the second involving a cube root and a product, and the third involving a cube root and an exponent. Each term requires a specific approach, utilizing the properties of roots and exponents to simplify them. By understanding the order of operations and the relationships between different mathematical functions, we can systematically reduce the complexity of the expression. This step-by-step breakdown is crucial for accuracy and clarity in the solution process. Let's begin by examining the first term and understanding how square roots and squares interact with each other.

First Term: (2)2\sqrt{(2)}^2

The first term of the expression is (2)2\sqrt{(2)}^2. This term involves a square root and an exponent. Specifically, it's the square root of 2, squared. A fundamental property in mathematics states that squaring a square root effectively cancels out the root, provided we are dealing with non-negative numbers. In simpler terms, the square root of a number, when squared, returns the original number. This is because the square root operation is the inverse of the squaring operation. Therefore, (2)2\sqrt{(2)}^2 simplifies directly to 2. This simplification is a direct application of the inverse relationship between square roots and squares. Understanding this relationship is key to simplifying many mathematical expressions. This initial simplification sets the stage for tackling the remaining terms in the expression. Now, let's move on to the second term and see how we can simplify it using the properties of cube roots and multiplication.

Second Term: 9×73\sqrt[3]{9 \times 7}

The second term is 9×73\sqrt[3]{9 \times 7}. This involves a cube root and a product. To simplify this, we first perform the multiplication inside the cube root. 9 multiplied by 7 equals 63. So, we now have 633\sqrt[3]{63}. The next step is to determine if 63 has any perfect cube factors. A perfect cube is a number that can be obtained by cubing an integer (e.g., 1, 8, 27, 64, etc.). In this case, 63 does not have any perfect cube factors other than 1. Its prime factorization is 32×73^2 \times 7, and since none of the prime factors appear three times, we cannot simplify the cube root further. Therefore, 9×73\sqrt[3]{9 \times 7} simplifies to 633\sqrt[3]{63}. While we couldn't completely eliminate the cube root, understanding the process of factorization and identifying perfect cube factors is crucial in simplifying such expressions. Now, let's tackle the third and final term, which involves a cube root and an exponent, to complete our simplification journey.

Third Term: (2)63\sqrt[3]{(2)^6}

The third term is (2)63\sqrt[3]{(2)^6}. This term involves a cube root and an exponent. Here, we have the cube root of 2 raised to the power of 6. To simplify this, we can use the property that amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}. Applying this property, we can rewrite the term as 2632^{\frac{6}{3}}. The exponent 63\frac{6}{3} simplifies to 2. Therefore, the expression becomes 222^2, which equals 4. So, (2)63\sqrt[3]{(2)^6} simplifies to 4. This simplification showcases the power of understanding the relationship between roots and exponents. By converting the cube root into a fractional exponent, we were able to easily simplify the term. This completes the individual simplification of each term in the expression. Now, let's combine these simplified terms to arrive at the final answer.

Combining the Simplified Terms

Now that we have simplified each term individually, we can combine them to find the final answer. We found that: 1. (2)2=2\sqrt{(2)}^2 = 2 2. 9×73=633\sqrt[3]{9 \times 7} = \sqrt[3]{63} 3. (2)63=4\sqrt[3]{(2)^6} = 4 The original expression was (2)2+9×73−(2)63\sqrt{(2)}^2+\sqrt[3]{9 \times 7}-\sqrt[3]{(2)^6}. Substituting the simplified values, we get: 2+633−42 + \sqrt[3]{63} - 4. Combining the constants, we have 2−4=−22 - 4 = -2. Therefore, the expression simplifies to −2+633-2 + \sqrt[3]{63}. This is the most simplified form of the expression, as 633\sqrt[3]{63} cannot be simplified further. The final result is a combination of an integer and a cube root, demonstrating the interplay between different types of numbers in mathematical expressions. In this final step, the importance of accurate substitution and combination of terms is highlighted, leading us to the ultimate solution.

Final Answer

After breaking down the expression (2)2+9×73−(2)63\sqrt{(2)}^2+\sqrt[3]{9 \times 7}-\sqrt[3]{(2)^6} into its individual components, simplifying each term using mathematical properties, and then combining the results, we arrive at the final answer. The simplified form of the expression is −2+633-2 + \sqrt[3]{63}. This result showcases the elegance of mathematical simplification, where a seemingly complex expression can be reduced to a more manageable and understandable form. The process involved understanding the relationships between square roots, cube roots, exponents, and basic arithmetic operations. Each step built upon the previous one, demonstrating the sequential nature of mathematical problem-solving. This final answer not only provides the numerical solution but also reinforces the importance of methodical and accurate application of mathematical principles. With this solution, we conclude our exploration of this mathematical expression, highlighting the beauty and precision inherent in mathematics.

Conclusion

In conclusion, the journey through the expression (2)2+9×73−(2)63\sqrt{(2)}^2+\sqrt[3]{9 \times 7}-\sqrt[3]{(2)^6} has been a testament to the power of mathematical simplification and the interconnectedness of various mathematical concepts. We began with a complex-looking expression and, through a step-by-step process, dissected it into manageable parts. We applied the fundamental properties of square roots, cube roots, and exponents, and we combined these with basic arithmetic operations to arrive at the simplified form: −2+633-2 + \sqrt[3]{63}. This process has underscored the importance of understanding the underlying principles of mathematics, such as the inverse relationship between squaring and square rooting, and the conversion of roots to fractional exponents. Furthermore, it has highlighted the significance of methodical problem-solving, where each step is carefully executed to ensure accuracy and clarity. The final result is not just a numerical answer; it's a culmination of a logical and systematic approach to problem-solving, which is a skill that extends far beyond the realm of mathematics. This exercise serves as a valuable reminder of the elegance and precision that mathematics offers, and the satisfaction that comes from successfully navigating its complexities.