Step-by-Step Solution Evaluating (21/4)^3 + (8/2)^2 - √(4/16) * (8/2 + 3/6)

Introduction

In this detailed guide, we will walk through the step-by-step evaluation of the mathematical expression ${\left(\frac{21}{4}\right)^3+\left(\frac{8}{2}\right)^2-\sqrt{\frac{4}{16}}\left(\frac{8}{2}+\frac{3}{6}\right)}$. This expression combines several arithmetic operations, including exponentiation, division, square roots, addition, and subtraction. By breaking down each part of the expression, we can systematically arrive at the final answer. This guide is designed to provide a clear and thorough understanding of the process, making it accessible for learners of all levels. This mathematical exploration aims not only to provide the solution but also to enhance your problem-solving skills in mathematics.

Our journey through this problem will highlight the importance of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We will also delve into simplifying fractions and radicals, crucial techniques in mathematical manipulations. Understanding these concepts is essential for tackling more complex problems in algebra and calculus. The step-by-step approach ensures that each operation is performed correctly, building a strong foundation for more advanced mathematical concepts. This comprehensive guide aims to equip you with the skills necessary to confidently approach similar problems in the future, fostering a deeper appreciation for mathematical precision and problem-solving strategies. The detailed explanation will serve as a valuable resource for students, educators, and anyone interested in honing their mathematical abilities.

Step 1: Evaluating the Exponents

The first step in evaluating this expression is to address the exponents. We have two terms with exponents: ${\left(\frac{21}{4}\right)^3}$ and ${\left(\frac{8}{2}\right)^2}$. Let's evaluate each term separately.

Evaluating ${\left(\frac{21}{4}\right)^3}$

To evaluate ${\left(\frac{21}{4}\right)^3}$, we need to raise both the numerator and the denominator to the power of 3. This means we calculate ${21^3}$ and ${4^3}$ separately and then divide the results. The process of evaluating exponents is fundamental in mathematics, serving as the cornerstone for more advanced algebraic and calculus operations. A clear understanding of how to handle exponents is crucial for solving equations, simplifying expressions, and grasping the concepts behind exponential functions and growth models. The ability to accurately compute exponents ensures that we can navigate complex mathematical terrains with precision and confidence.

First, let's calculate ${21^3}$, which is ${21 \times 21 \times 21}$.

• ${21 \times 21 = 441}$
• ${441 \times 21 = 9261}$

So, ${21^3 = 9261}$. Next, we calculate ${4^3}$, which is ${4 \times 4 \times 4}$.

• ${4 \times 4 = 16}$
• ${16 \times 4 = 64}$

Thus, ${4^3 = 64}$. Therefore, ${\left(\frac{21}{4}\right)^3 = \frac{9261}{64}}$. This result is a crucial step in our overall evaluation, setting the stage for further arithmetic operations. The accurate calculation of this exponential term ensures the integrity of our subsequent calculations and final answer. It's important to meticulously perform each step to maintain the precision required in mathematical problem-solving. This calculated value will now be used in the broader context of the expression, where it will be combined with other terms to reach the final solution.

Evaluating ${\left(\frac{8}{2}\right)^2}$

Next, we evaluate ${\left(\frac{8}{2}\right)^2}$. Before raising to the power, we can simplify the fraction ${\frac{8}{2}}$, which equals 4. So, the term becomes ${4^2}$.

Now, we calculate ${4^2}$, which is ${4 \times 4 = 16}$. Therefore, ${\left(\frac{8}{2}\right)^2 = 16}$. Simplifying the fraction before applying the exponent greatly simplifies the calculation process, reducing the chance of errors. This step highlights the importance of looking for opportunities to simplify expressions before diving into complex operations. The ability to simplify fractions and expressions is a fundamental skill in mathematics, enabling us to handle more intricate problems with ease and efficiency. This result, a straightforward integer, will now be combined with other terms, contributing to the final solution of the expression. The simplification process not only makes the calculation easier but also provides a clear, manageable value to work with.

Step 2: Evaluating the Square Root

Now, let's evaluate the square root term: ${\sqrt{\frac{4}{16}}}$ . To simplify this, we first look at the fraction inside the square root, ${\frac{4}{16}}$, which can be simplified to ${\frac{1}{4}}$. The simplification of fractions is a vital skill in mathematical operations, enabling us to handle expressions more efficiently and accurately. By reducing fractions to their simplest forms, we minimize the risk of computational errors and streamline the problem-solving process. This skill is not just limited to basic arithmetic but extends into more advanced topics like algebra, calculus, and beyond, where complex fractions are frequently encountered. Simplifying the fraction under the square root significantly eases the subsequent calculation, providing a more manageable value to work with.

So, we have ${\sqrt{\frac{1}{4}}}$ . Taking the square root of both the numerator and the denominator separately, we get ${\frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}}$. The ability to extract the square root of fractions is a crucial mathematical skill, particularly useful in various scientific and engineering applications. By understanding how to handle square roots, we can solve equations, simplify expressions, and tackle problems involving geometric shapes and spatial relationships. Extracting square roots involves breaking down the number into its prime factors and identifying perfect squares, a process that reinforces our understanding of number theory and algebraic manipulations. This step ensures that we are working with a simplified and exact value, which is critical for the precision of the final answer.

Step 3: Evaluating the Parenthetical Term

Next, we need to evaluate the expression inside the parentheses: ${\left(\frac{8}{2}+\frac{3}{6}\right)}$. First, we simplify each fraction individually.

• ${\frac{8}{2}}$ simplifies to 4.
• ${\frac{3}{6}}$ simplifies to ${\frac{1}{2}}$.

Now, we add the simplified fractions: ${4 + \frac{1}{2}}$. To add these, we convert 4 to a fraction with a denominator of 2, which is ${\frac{8}{2}}$. Then, we add the fractions:

${ \frac{8}{2} + \frac{1}{2} = \frac{8+1}{2} = \frac{9}{2} }$

Thus, ${\left(\frac{8}{2}+\frac{3}{6}\right) = \frac{9}{2}}$. Parenthetical terms serve as crucial boundaries in mathematical expressions, dictating the order of operations and ensuring that specific calculations are performed before others. Evaluating parenthetical expressions is a fundamental skill, reinforcing our understanding of the order of operations and the importance of adhering to established mathematical conventions. The correct handling of parentheses is essential for solving complex equations, simplifying algebraic expressions, and tackling advanced mathematical problems. The result of this parenthetical evaluation is now a single, simplified fraction, which will be used in the subsequent steps of the calculation to determine the overall value of the expression.

Step 4: Performing Multiplication

Now, we multiply the results from Step 2 and Step 3: ${\frac{1}{2} \times \frac{9}{2}}$. Multiplying fractions involves multiplying the numerators and the denominators separately.

${ \frac{1}{2} \times \frac{9}{2} = \frac{1 \times 9}{2 \times 2} = \frac{9}{4} }$

So, ${\sqrt{\frac{4}{16}}\left(\frac{8}{2}+\frac{3}{6}\right) = \frac{9}{4}}$. Multiplication of fractions is a key arithmetic skill, vital for simplifying expressions, solving equations, and tackling a wide array of mathematical problems. Understanding how to multiply fractions efficiently is essential in various fields, including mathematics, physics, engineering, and finance. The process of multiplying fractions involves a clear and straightforward application of mathematical rules, enhancing our ability to perform calculations accurately and confidently. This simplified value will now be used in the final step, where it will be subtracted from the sum of the exponential terms, ultimately leading to the final solution of the expression.

Step 5: Combining All Terms

Finally, we combine all the terms we've calculated:

${ \left(\frac{21}{4}\right)^3+\left(\frac{8}{2}\right)^2-\sqrt{\frac{4}{16}}\left(\frac{8}{2}+\frac{3}{6}\right) = \frac{9261}{64} + 16 - \frac{9}{4} }$

To combine these terms, we need a common denominator. The least common denominator (LCD) of 64, 1, and 4 is 64. So, we convert each term to have a denominator of 64.

• ${\frac{9261}{64}}$ remains the same.
• ${16}$ becomes ${\frac{16 \times 64}{64} = \frac{1024}{64}}$.
• ${\frac{9}{4}}$ becomes ${\frac{9 \times 16}{4 \times 16} = \frac{144}{64}}$.

Now we can add and subtract the fractions:

${ \frac{9261}{64} + \frac{1024}{64} - \frac{144}{64} = \frac{9261 + 1024 - 144}{64} = \frac{10141}{64} }$

So, the final result is ${\frac{10141}{64}}$. Combining terms with a common denominator is a fundamental arithmetic skill, crucial for simplifying expressions and solving equations in algebra and beyond. Understanding how to find the least common denominator (LCD) and convert fractions to equivalent forms is essential for performing addition and subtraction accurately. This skill is not only important in academic mathematics but also has practical applications in various fields, including engineering, physics, and finance, where complex calculations often involve fractions and mixed numbers. The final result, a single simplified fraction, represents the culmination of all the previous steps, providing a concise and accurate solution to the original mathematical expression.

Conclusion

In conclusion, the value of the expression ${\left(\frac{21}{4}\right)^3+\left(\frac{8}{2}\right)^2-\sqrt{\frac{4}{16}}\left(\frac{8}{2}+\frac{3}{6}\right)}$ is ${\frac{10141}{64}}$. We arrived at this result by systematically evaluating each part of the expression, following the order of operations, and simplifying fractions and square roots. This detailed walkthrough not only provides the solution but also highlights the importance of each step in the process. Understanding these concepts is crucial for tackling more complex problems in mathematics. The step-by-step approach used in this guide underscores the significance of methodical problem-solving, reinforcing key arithmetic and algebraic principles. This thorough exploration serves as a valuable resource for students, educators, and anyone seeking to enhance their mathematical skills. This comprehensive solution demonstrates the power of breaking down complex problems into manageable parts, ultimately leading to an accurate and understandable result. The final answer, ${\frac{10141}{64}}$, is a testament to the precision and clarity achieved through careful mathematical manipulation and a solid understanding of fundamental operations.