Calculating The Value Of X In The Expression X_7 = 7^3 + (2 × 7) + (3 × 7^0)

In this article, we will delve into the mathematical expression x_7 = 7^3 + (2 × 7) + (3 × 7^0) and determine the value of x. This problem involves understanding the order of operations, exponents, and basic arithmetic. Let's break down the calculation step by step to arrive at the correct answer. Understanding such expressions is crucial for anyone delving into mathematics, especially in fields like computer science where different number systems are frequently encountered. The ability to correctly evaluate these expressions ensures a solid foundation for more complex mathematical concepts. Our focus will be on providing a clear, step-by-step guide that will enable you to solve similar problems with ease. We aim to not only give the answer but also to explain the underlying principles and methodologies involved.

The given expression is x_7 = 7^3 + (2 × 7) + (3 × 7^0). This expression consists of several terms, each involving powers of 7 and simple arithmetic operations. To solve it correctly, we need to understand the order of operations (PEMDAS/BODMAS), which dictates that we should first handle parentheses, then exponents, followed by multiplication and division, and finally addition and subtraction. Let's break down each term individually.

The first term is 7^3, which means 7 raised to the power of 3, or 7 multiplied by itself three times (7 × 7 × 7). The second term is (2 × 7), a simple multiplication. The third term involves 7^0. Any non-zero number raised to the power of 0 is 1, so 7^0 equals 1. This might seem counterintuitive at first, but it's a fundamental rule in mathematics that ensures consistency across various mathematical operations and expressions. Understanding this rule is crucial for simplifying and solving equations involving exponents. The fact that any non-zero number to the power of zero equals one is a cornerstone in algebra and calculus, ensuring smooth transitions in complex formulas and calculations. The interplay between exponents and arithmetic is a key aspect of mathematical problem-solving, often requiring careful attention to detail and a firm grasp of basic principles. As we continue, we will see how these individual components come together to form the final solution.

Now, let's calculate each term in the expression x_7 = 7^3 + (2 × 7) + (3 × 7^0) step by step.

1. Calculate 7^3: 7^3 means 7 multiplied by itself three times, which is 7 × 7 × 7. 7 × 7 equals 49, and then 49 × 7 equals 343. So, 7^3 = 343. This step is fundamental as it deals with exponentiation, a common operation in many mathematical problems. Exponentiation is not just a mathematical abstraction; it appears in real-world scenarios such as compound interest calculations, population growth models, and even in physics when calculating areas and volumes. The result, 343, becomes a significant component of our final answer, and understanding how we arrived at this value is critical for mastering exponentiation. Accuracy in this step is paramount because an error here will propagate through the rest of the calculation, leading to an incorrect final answer.
2. Calculate (2 × 7): This is a simple multiplication. 2 multiplied by 7 equals 14. Basic multiplication like this forms the building blocks of more complex arithmetic. Mastering multiplication tables and techniques enables quick and accurate calculations, essential in various mathematical contexts. The clarity of this step also underscores the importance of understanding fundamental operations. Multiplication is one of the four basic operations in arithmetic (addition, subtraction, multiplication, and division), and proficiency in multiplication is critical for more advanced math, including algebra and calculus.
3. Calculate (3 × 7^0): First, we need to evaluate 7^0. As mentioned earlier, any non-zero number raised to the power of 0 is 1. Therefore, 7^0 = 1. Next, we multiply 3 by 1, which equals 3. This step highlights a crucial mathematical rule – anything to the power of zero equals one. This rule often causes confusion but is fundamental for the consistency of mathematical systems. Applying this rule correctly ensures that our calculations align with established mathematical principles. Recognizing and applying such rules is a key skill in problem-solving, particularly in algebraic manipulations and equation solving.

Now that we have calculated each term separately, we can combine them to find the value of x_7. The expression is x_7 = 7^3 + (2 × 7) + (3 × 7^0). We found that 7^3 = 343, (2 × 7) = 14, and (3 × 7^0) = 3. So, we can rewrite the expression as x_7 = 343 + 14 + 3. Adding these values together, we get 343 + 14 = 357, and then 357 + 3 = 360. Therefore, x_7 = 360.

This final step brings together all the individual calculations we've performed, showcasing the importance of a systematic approach in problem-solving. By breaking down a complex expression into smaller, manageable parts, we simplify the process and minimize the risk of errors. Addition is one of the fundamental operations, and its proper application is critical for arriving at the correct final result. Accuracy in addition, just like in previous steps, is paramount. This final combination underscores the interconnectedness of mathematical operations – each step builds upon the previous one, culminating in a definitive answer.

So, the value of x_7 in the expression 7^3 + (2 × 7) + (3 × 7^0) is 360. Therefore, none of the provided options (A) 23, (B) 123, (C) 321, (D) 1023 are correct. The correct answer is 360. This result demonstrates the importance of careful calculation and attention to detail in mathematical problem-solving. It also underscores the need to double-check work, especially in scenarios where the calculated answer doesn't match the given options. Our step-by-step breakdown allows us to confidently assert the correctness of our solution. Furthermore, this exercise emphasizes the practical application of mathematical principles in everyday problem-solving, showing that math is not just an abstract subject but a tool for accurately analyzing and interpreting the world around us. The ability to arrive at a correct answer through systematic calculation is a valuable skill, not just in mathematics but in various aspects of life and work.

In conclusion, by carefully evaluating each term in the expression x_7 = 7^3 + (2 × 7) + (3 × 7^0), we determined that x_7 equals 360. This process involved understanding and applying the order of operations, evaluating exponents, and performing basic arithmetic. While the provided options did not include the correct answer, our step-by-step approach ensures the accuracy of our result. This exercise highlights the importance of methodical problem-solving and the fundamental role of arithmetic and exponential operations in mathematics. Ultimately, the ability to tackle such problems enhances one's mathematical proficiency and builds a solid foundation for more advanced topics. The systematic methodology we've employed here can be applied to a wide array of mathematical challenges, reinforcing the value of a clear, structured approach in problem-solving. This experience underscores that math, at its core, is a logical and precise discipline, where each step contributes to a coherent and verifiable final answer.