Finding The Value Of N In Scientific Notation Product

In the realm of mathematics, particularly when dealing with extremely large or small numbers, scientific notation becomes an indispensable tool. It provides a concise and manageable way to express these numbers, making calculations and comparisons significantly easier. This article delves into a problem involving scientific notation, where we aim to find the value of an unknown number n given the product of $1.3 imes 10^{-4}$ and n equals $2.6 imes 10^{12}$. Understanding how to manipulate numbers in scientific notation is crucial for various scientific and engineering applications, and this problem serves as an excellent illustration of the principles involved.

Before we tackle the problem at hand, it's essential to grasp the fundamentals of scientific notation. A number in scientific notation is expressed in the form a × 10^b, where a is a number between 1 and 10 (the coefficient or significand), and b is an integer (the exponent). The exponent indicates the number of places the decimal point must be moved to obtain the standard form of the number. A positive exponent signifies a large number, while a negative exponent indicates a small number.

For instance, the number 3,000,000 can be written in scientific notation as $3 imes 10^6$, where 3 is the coefficient and 6 is the exponent. Similarly, the number 0.000005 can be expressed as $5 imes 10^{-6}$. Scientific notation not only simplifies the representation of numbers but also streamlines arithmetic operations such as multiplication and division.

The problem we are addressing is as follows: The product of $1.3 imes 10^{-4}$ and a number n results in $2.6 imes 10^{12}$. Our objective is to determine the value of n. This problem exemplifies a common scenario in scientific calculations where we need to isolate an unknown variable within an equation involving numbers expressed in scientific notation. To solve this, we will employ the principles of algebraic manipulation and the rules governing exponents.

To find the value of n, we need to isolate it in the equation. The given equation is:

$(1.3 imes 10^{-4}) imes n = 2.6 imes 10^{12}$

To isolate n, we divide both sides of the equation by $1.3 imes 10^{-4}$:

n = rac{2.6 imes 10^{12}}{1.3 imes 10^{-4}}

Now, we can separate the coefficients and the powers of 10:

n = rac{2.6}{1.3} imes rac{10^{12}}{10^{-4}}

Dividing the coefficients, we get:

rac{2.6}{1.3} = 2

For the powers of 10, we use the rule of exponents which states that when dividing powers with the same base, we subtract the exponents:

rac{10^{12}}{10^{-4}} = 10^{12 - (-4)} = 10^{12 + 4} = 10^{16}

Therefore, the value of n is:

$n = 2 imes 10^{16}$

This result demonstrates how scientific notation simplifies the process of dividing very large and very small numbers. By separating the coefficients and the exponents, we can perform the division more easily and accurately.

We are provided with the following answer options:

A. $2 imes 10^{-8}$ B. $2 imes 10^{-3}$ C. $2 imes 10^{16}$ D. $2 imes 10^{48}$

Comparing our calculated value of n with the given options, we can see that the correct answer is:

C. $2 imes 10^{16}$

This confirms our solution and highlights the importance of careful calculation and attention to the rules of exponents when working with scientific notation.

In this article, we have successfully determined the value of n in the equation $(1.3 imes 10^{-4}) imes n = 2.6 imes 10^{12}$. By understanding and applying the principles of scientific notation and the rules of exponents, we found that n equals $2 imes 10^{16}$. This problem illustrates the practical application of scientific notation in simplifying calculations involving very large and very small numbers. Mastering these concepts is essential for anyone working in fields that require quantitative analysis and problem-solving.

The ability to manipulate numbers in scientific notation is a fundamental skill in mathematics and science. It allows us to express and work with numbers that would otherwise be cumbersome to handle. The steps involved in solving this problem, from isolating the unknown variable to applying the rules of exponents, are crucial techniques that can be applied to a wide range of mathematical and scientific challenges. Whether you are a student learning the basics of scientific notation or a professional applying these concepts in your work, a solid understanding of these principles will undoubtedly prove invaluable.

Scientific notation is not merely a mathematical concept; it is a practical tool with numerous applications across various fields. In physics, for example, it is used to express quantities such as the speed of light ($3 imes 10^8$ meters per second) and the gravitational constant ($6.674 imes 10^{-11}$ N⋅m²/kg²). In chemistry, it is essential for representing Avogadro's number ($6.022 imes 10^{23}$ particles per mole) and the sizes of atoms and molecules. Astronomy relies heavily on scientific notation to express the vast distances between celestial bodies and the masses of stars and planets.

In computer science, scientific notation can be used to represent the storage capacity of memory devices or the speed of data transmission. Engineering disciplines also utilize scientific notation in calculations involving large structures, electrical circuits, and fluid dynamics. The ability to work with scientific notation is thus a versatile skill that transcends specific academic subjects and professional domains. It is a key component of quantitative literacy and critical thinking.

Moreover, scientific notation plays a crucial role in data analysis and statistics. When dealing with large datasets, it is often necessary to represent numbers in a compact and easily interpretable form. Scientific notation allows researchers to focus on the significant digits of a number while avoiding the clutter of many trailing zeros. This is particularly important when presenting results in scientific publications or reports, where clarity and precision are paramount. Statistical software and programming languages often provide built-in functions for working with scientific notation, further highlighting its importance in modern data-driven fields.

To effectively work with scientific notation, there are several key strategies to keep in mind. First and foremost, it is essential to understand the basic form of scientific notation, which is a × 10^b, where 1 ≤ |a| < 10 and b is an integer. This ensures that the number is represented in a standardized format, making it easier to compare and perform calculations.

When multiplying numbers in scientific notation, multiply the coefficients and add the exponents. For example, $(2 imes 10^3) imes (3 imes 10^4) = (2 imes 3) imes 10^{3+4} = 6 imes 10^7$. Conversely, when dividing numbers in scientific notation, divide the coefficients and subtract the exponents. For instance, $(6 imes 10^7) / (2 imes 10^3) = (6 / 2) imes 10^{7-3} = 3 imes 10^4$.

When adding or subtracting numbers in scientific notation, it is necessary to ensure that the exponents are the same. If the exponents are different, adjust one of the numbers so that they match. For example, to add $3 imes 10^4$ and $2 imes 10^3$, convert $2 imes 10^3$ to $0.2 imes 10^4$ and then add the coefficients: $3 imes 10^4 + 0.2 imes 10^4 = 3.2 imes 10^4$.

Finally, always double-check your calculations and ensure that the final answer is in the correct form of scientific notation. This involves verifying that the coefficient is between 1 and 10 and that the exponent is an integer. Practicing these techniques regularly will help solidify your understanding of scientific notation and improve your ability to work with it effectively.

In conclusion, mastering scientific notation is a crucial skill for anyone engaging with quantitative disciplines. From simplifying complex calculations to representing vast or minuscule values, its applications are far-reaching. The ability to manipulate numbers in this format not only enhances mathematical proficiency but also fosters a deeper appreciation for the scale of the universe and the precision required in scientific endeavors. By understanding the fundamental principles and practicing regularly, individuals can confidently navigate the challenges posed by large and small numbers, unlocking new possibilities in their academic and professional pursuits. Whether in physics, chemistry, astronomy, computer science, or any other field that relies on quantitative analysis, the mastery of scientific notation stands as a testament to one's preparedness and aptitude.