Solving (4.8 X 10^-7) / (1.6 X 10^-11) A Step-by-Step Guide

In the realm of mathematics and scientific calculations, scientific notation serves as a powerful tool for expressing extremely large or small numbers concisely. This article dives deep into the process of dividing numbers expressed in scientific notation, specifically addressing the problem: ${\frac{4.8 \times 10^{-7}}{1.6 \times 10^{-11}}}$. We will not only solve this particular problem step-by-step but also explore the underlying principles and techniques involved in such calculations. Understanding these concepts is crucial for anyone working with scientific data, engineering problems, or advanced mathematical concepts. So, let's embark on this journey to demystify scientific notation and master the art of division within this framework.

Decoding Scientific Notation: A Foundation for Division

Before we tackle the division problem, it's essential to grasp the fundamentals of scientific notation. Scientific notation, also known as standard form, is a way of writing numbers as the product of two factors: a coefficient and a power of 10. The coefficient is a number between 1 (inclusive) and 10 (exclusive), and the power of 10 indicates the magnitude of the number. For instance, the number 3,000,000 can be expressed in scientific notation as ${3 \times 10^6}$, where 3 is the coefficient and ${10^6}$ represents one million. Similarly, a small number like 0.000005 can be written as ${5 \times 10^{-6}}$.

This notation system is particularly useful when dealing with numbers that have many digits or are very close to zero. It simplifies calculations, reduces the risk of errors, and makes it easier to compare numbers of different magnitudes. The beauty of scientific notation lies in its ability to condense information and provide a clear representation of a number's scale. In our problem, we are dealing with numbers in scientific notation, making it imperative to understand the rules governing their division. When dividing numbers in scientific notation, we essentially divide the coefficients and subtract the exponents of the powers of 10. This principle forms the backbone of our solution strategy.

Step-by-Step Solution: Dividing Numbers in Scientific Notation

Now, let's apply our understanding of scientific notation to solve the problem: ${\frac{4.8 \times 10^{-7}}{1.6 \times 10^{-11}}}$. This equation might seem daunting at first glance, but breaking it down into smaller, manageable steps will reveal its simplicity. The key to success lies in treating the coefficients and the powers of 10 separately. This approach not only simplifies the calculation but also minimizes the chances of making errors. By isolating the components, we can focus on each part individually and then combine the results to arrive at the final answer. This methodical process is a cornerstone of problem-solving in mathematics and science.

Step 1: Divide the Coefficients

The first step involves dividing the coefficients: 4.8 and 1.6. When you divide 4.8 by 1.6, you get 3. This can be expressed as:

{rac{4.8}{1.6} = 3}

This step is straightforward and forms the foundation of our calculation. The result, 3, will be the coefficient of our answer in scientific notation. It's important to note that this coefficient falls within the acceptable range for scientific notation, which is between 1 and 10. If the result were outside this range, we would need to adjust it accordingly. However, in this case, the coefficient is perfectly aligned with the requirements of scientific notation, making our task smoother.

Step 2: Divide the Powers of 10

Next, we tackle the powers of 10. We have ${10^{-7}}$ divided by ${10^{-11}}$. When dividing exponential terms with the same base, you subtract the exponents. In this case, we subtract -11 from -7:

${10^{-7} \div 10^{-11} = 10^{-7 - (-11)} = 10^{-7 + 11} = 10^4}$

This step is crucial in determining the magnitude of our final answer. By subtracting the exponents, we effectively account for the difference in scale between the two numbers. The result, ${10^4}$, indicates that our answer will be a number in the ten-thousands range. This exponent manipulation is a fundamental skill in working with scientific notation and is widely applicable in various scientific and engineering contexts.

Step 3: Combine the Results

Now, we combine the results from Step 1 and Step 2. We have the coefficient 3 and the power of 10, ${10^4}$. Multiplying these together gives us:

${3 \times 10^4}$

This is our final answer in scientific notation. It represents the result of dividing ${4.8 \times 10^{-7}}$ by ${1.6 \times 10^{-11}}$. The answer, ${3 \times 10^4}$, or 30,000, provides a clear and concise representation of the result. This final step underscores the elegance of scientific notation in simplifying complex calculations and presenting numbers in an easily understandable format.

Identifying the Correct Answer: Multiple Choice Analysis

In a multiple-choice setting, the ability to quickly and accurately identify the correct answer is paramount. Having solved the problem, we know the answer is ${3 \times 10^4}$. Now, let's examine the given options:

A. ${3.0 \times 10^4}$ B. ${3.0 \times 10^{-4}}$ C. ${3.0 \times 10^{-18}}$ D. ${3.2 \times 10^{18}}$ E. ${3.2 \times 10^4}$

By comparing our solution with the options, we can clearly see that option A, ${3.0 \times 10^4}$, matches our calculated result. The other options either have the wrong exponent or the wrong coefficient. This process of elimination is a valuable strategy in multiple-choice exams, allowing you to narrow down the possibilities and increase your chances of selecting the correct answer. The ability to confidently arrive at the solution and then match it with the appropriate option is a testament to a strong understanding of the underlying concepts.

Common Pitfalls and How to Avoid Them

Working with scientific notation can sometimes be tricky, and there are common pitfalls that students often encounter. Being aware of these potential errors can help you avoid them and ensure accuracy in your calculations. One frequent mistake is incorrectly subtracting the exponents when dividing powers of 10. Remember, when dividing, you subtract the exponent in the denominator from the exponent in the numerator. For example, dividing ${10^{-7}}$ by ${10^{-11}}$ requires subtracting -11 from -7, which gives you ${10^4}$, not ${10^{-18}}$.

Another common error is forgetting to adjust the coefficient if it falls outside the range of 1 to 10. If, after dividing, the coefficient is less than 1 or greater than or equal to 10, you need to adjust it and modify the exponent accordingly. For instance, if you end up with ${0.5 \times 10^5}$, you should rewrite it as ${5 \times 10^4}$. Similarly, ${12 \times 10^3}$ should be adjusted to ${1.2 \times 10^4}$. Paying close attention to these details can significantly improve your accuracy when working with scientific notation.

Real-World Applications: The Importance of Scientific Notation

Scientific notation is not just a mathematical concept; it has widespread applications in various scientific and engineering disciplines. In physics, it is used to express extremely large numbers like the speed of light (approximately ${3 \times 10^8}$ meters per second) or very small numbers like the mass of an electron (approximately ${9.11 \times 10^{-31}}$ kilograms). In chemistry, scientific notation is essential for representing concentrations of solutions, Avogadro's number (approximately ${6.022 \times 10^{23}}$ particles per mole), and other fundamental constants.

Astronomy relies heavily on scientific notation to express distances between celestial objects, the masses of stars, and the ages of galaxies. Engineering fields, such as electrical engineering and materials science, also utilize scientific notation to handle very large and very small quantities. The ability to work comfortably with scientific notation is therefore a crucial skill for anyone pursuing a career in science, technology, engineering, or mathematics (STEM). Its importance extends beyond academic settings and into practical, real-world applications where precision and efficiency are paramount.

Mastering Scientific Notation: Practice Makes Perfect

In conclusion, dividing numbers in scientific notation involves dividing the coefficients and subtracting the exponents of the powers of 10. By following a step-by-step approach and paying attention to potential pitfalls, you can confidently solve these types of problems. Understanding scientific notation is not just about performing calculations; it's about grasping the scale of numbers and appreciating the elegance of this notation system. To truly master scientific notation, practice is key. Work through various examples, challenge yourself with complex problems, and apply these concepts in different contexts. The more you practice, the more comfortable and proficient you will become, unlocking the full potential of scientific notation in your scientific and mathematical endeavors.