Solving The Mathematical Expression (1/2) * (4/3) * 3.14 * (7.8)^3 + (3.14)(7.8)^2(8.7)

Introduction to Mathematical Expressions

In this comprehensive guide, we will explore the intricacies of evaluating the mathematical expression: ${\frac{1}{2} \cdot \frac{4}{3} \cdot 3.14 \cdot(7.8)^3+(3.14)(7.8)^2(8.7)}$. This expression combines several mathematical operations, including multiplication, exponentiation, and addition, making it a valuable exercise for enhancing your mathematical skills. Understanding the order of operations and how to apply them correctly is crucial for arriving at the correct solution. This article breaks down each step involved in calculating the expression, providing detailed explanations and insights along the way.

Breaking Down the Expression

To effectively tackle this expression, we need to understand its components and the order in which the operations should be performed. The expression consists of two main terms, separated by an addition sign: ${\frac{1}{2} \cdot \frac{4}{3} \cdot 3.14 \cdot(7.8)^3}$ and ${(3.14)(7.8)^2(8.7)}$. According to the order of operations (PEMDAS/BODMAS), we must first address any parentheses or exponents, followed by multiplication and division, and finally, addition and subtraction. Let's begin by evaluating the first term, which includes exponentiation and multiplication.

Step-by-Step Evaluation

1. Calculate the Exponent: The first step involves calculating ${(7.8)^3}$, which means 7.8 raised to the power of 3. This is equivalent to 7.8 multiplied by itself three times: ${7.8 \times 7.8 \times 7.8}$. This calculation will give us the value of the exponent term, which is essential for the subsequent multiplication operations.
2. Multiply the Fractions and Constant: Next, we multiply the fractions ${\frac{1}{2}}$ and ${\frac{4}{3}}$ with the constant 3.14. This involves multiplying the numerators together and the denominators together, and then multiplying the result by 3.14. Understanding fraction multiplication is key to simplifying this part of the expression.
3. Multiply the Results: We then multiply the result from the exponentiation step with the result from the multiplication of fractions and the constant. This step combines the outcomes of the previous calculations to form the first term of the expression. Careful multiplication is crucial here to avoid errors and ensure an accurate result.
4. Calculate the Second Term: Now, we shift our focus to the second term of the expression, ${(3.14)(7.8)^2(8.7)}$. This term also involves exponentiation and multiplication. We start by calculating ${(7.8)^2}$, which is 7.8 raised to the power of 2, or 7.8 multiplied by itself. This result is then multiplied by 3.14 and 8.7.
5. Perform the Final Addition: Finally, we add the results obtained from evaluating the first and second terms. This addition combines the two main components of the expression, leading us to the final answer. Ensuring accuracy in this final step is paramount for the overall correctness of the solution.

Detailed Breakdown of Each Term

Evaluating the First Term: {\frac{1}{2} \cdot rac{4}{3} \cdot 3.14 \cdot(7.8)^3}

In this section, we delve deeper into the step-by-step evaluation of the first term: {\frac{1}{2} \cdot rac{4}{3} \cdot 3.14 \cdot(7.8)^3}. We will break down each operation, providing detailed calculations and explanations to ensure clarity and understanding. This term combines fractions, a constant, and an exponent, making it a comprehensive exercise in mathematical operations. Grasping how to accurately evaluate this term is crucial for solving the entire expression.

1. Calculating the Exponent: ${(7.8)^3}$

The first step in evaluating the first term is to calculate ${(7.8)^3}$. This means raising 7.8 to the power of 3, which is equivalent to multiplying 7.8 by itself three times: ${7.8 \times 7.8 \times 7.8}$. This is a fundamental operation in mathematics, and its accurate calculation is vital for the rest of the expression. We will perform this multiplication step by step to ensure precision.

• First, multiply ${7.8 \times 7.8}$: ${7.8 \times 7.8 = 60.84}$
• Next, multiply the result by 7.8: ${60.84 \times 7.8 = 474.552}$

Therefore, ${(7.8)^3 = 474.552}$. This result will be used in the subsequent multiplication steps. Ensuring the correct exponent calculation sets the stage for the accurate evaluation of the entire term.

2. Multiplying the Fractions and Constant: {\frac{1}{2} \cdot rac{4}{3} \cdot 3.14}

The next step involves multiplying the fractions ${\frac{1}{2}}$ and ${\frac{4}{3}}$ with the constant 3.14. This operation requires understanding how to multiply fractions and how to incorporate a decimal value into the multiplication. We will break this down into smaller steps to make it easier to follow.

• First, multiply the fractions {\frac{1}{2} \times rac{4}{3}}: To multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together: {\frac{1}{2} \times rac{4}{3} = rac{1 \times 4}{2 \times 3} = rac{4}{6}}
• We can simplify the fraction ${\frac{4}{6}}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2: {\frac{4}{6} = rac{4 \div 2}{6 \div 2} = rac{2}{3}}
• Next, multiply the simplified fraction ${\frac{2}{3}}$ by the constant 3.14: {\frac{2}{3} \times 3.14 = rac{2 \times 3.14}{3} = rac{6.28}{3}}
• Now, divide 6.28 by 3: ${\frac{6.28}{3} \approx 2.0933}$

Therefore, the result of multiplying the fractions and the constant is approximately 2.0933. This result will be used in the next multiplication step, where we combine it with the exponent result.

3. Multiplying the Results: ${2.0933 \cdot 474.552}$

Now that we have calculated the exponent and the product of the fractions and the constant, we need to multiply these two results together. This step combines the outcomes of the previous calculations to form the value of the first term. Accurate multiplication is essential here to ensure the correct final value.

• Multiply ${2.0933 \times 474.552}$: ${2.0933 \times 474.552 \approx 993.18}$

Therefore, the value of the first term {\frac{1}{2} \cdot rac{4}{3} \cdot 3.14 \cdot(7.8)^3} is approximately 993.18. This result is a significant component of the overall expression, and its accurate calculation is crucial for the final answer.

Evaluating the Second Term: ${(3.14)(7.8)^2(8.7)}$

Having thoroughly evaluated the first term, we now turn our attention to the second term in the expression: ${(3.14)(7.8)^2(8.7)}$. This term also involves exponentiation and multiplication, but it presents a slightly different combination of operations. We will break down this term into manageable steps, providing detailed calculations and explanations to ensure a clear understanding. Successfully evaluating this term is essential for obtaining the final solution to the entire expression.

1. Calculating the Exponent: ${(7.8)^2}$

The first step in evaluating the second term is to calculate ${(7.8)^2}$. This means raising 7.8 to the power of 2, which is equivalent to multiplying 7.8 by itself: ${7.8 \times 7.8}$. This is a fundamental mathematical operation, and its accurate calculation is crucial for the rest of the expression. We will perform this multiplication carefully to ensure precision.

• Multiply ${7.8 \times 7.8}$: ${7.8 \times 7.8 = 60.84}$

Therefore, ${(7.8)^2 = 60.84}$. This result will be used in the subsequent multiplication steps. Correct exponent calculation is the foundation for accurately evaluating the entire term.

2. Multiplying the Results: ${3.14 \cdot 60.84 \cdot 8.7}$

Now that we have calculated the exponent, we need to multiply the result by 3.14 and 8.7. This involves performing sequential multiplication operations, ensuring that each step is accurate. We will break this down into smaller multiplications to make it easier to follow and minimize errors.

• First, multiply ${3.14 \times 60.84}$: ${3.14 \times 60.84 = 191.0376}$
• Next, multiply the result by 8.7: ${191.0376 \times 8.7 \approx 1661.027}$

Therefore, the value of the second term ${(3.14)(7.8)^2(8.7)}$ is approximately 1661.027. This result is the second significant component of the overall expression, and its accurate calculation is crucial for the final answer.

Final Calculation and Conclusion

Performing the Final Addition

With both terms of the expression evaluated, the final step is to add them together. This will give us the final result of the entire expression: {\frac{1}{2} \cdot rac{4}{3} \cdot 3.14 \cdot(7.8)^3+(3.14)(7.8)^2(8.7)}. We will add the results obtained in the previous sections to arrive at the solution.

• Add the results of the first and second terms: ${993.18 + 1661.027 \approx 2654.207}$

Conclusion

The final result of the expression {\frac{1}{2} \cdot rac{4}{3} \cdot 3.14 \cdot(7.8)^3+(3.14)(7.8)^2(8.7)} is approximately 2654.207. This calculation involved a series of mathematical operations, including exponentiation, multiplication, and addition. By breaking down the expression into smaller, manageable steps, we were able to accurately evaluate each component and arrive at the final answer. Understanding the order of operations and performing each step with precision are key to solving such mathematical expressions.

This comprehensive guide has provided a detailed, step-by-step explanation of how to evaluate the given expression. By following these methods, you can confidently tackle similar mathematical problems and enhance your problem-solving skills. The process of breaking down complex problems into simpler steps is a valuable strategy in mathematics and many other fields.