Problem 68 - Beginning and Intermediate Algebra

Question

Convert to scientific notation. $$6.27 \times 10^{-10}$$

Step-by-Step Solution

Answer

The given number \(6.27 \times 10^{-10}\) is already in the correct scientific notation format. No conversion is needed.

Step 1: Check if the coefficient is between 1 and 10

In this case, the coefficient is 6.27, which is between 1 and 10.

Step 2: Check if the number is multiplied by a power of 10

We see that the number is multiplied by \(10^{-10}\), which is indeed a power of 10. Since the given number meets both requirements, it is already in scientific notation. So, there is no conversion needed:

Step 3: Final Answer

The scientific notation of \(6.27 \times 10^{-10}\) is already in the correct format: \(6.27 \times 10^{-10}\)

Key Concepts

Mathematical Notation

Mathematical notation is a universal language used to express mathematical ideas unambiguously. It encompasses symbols and expressions that help convey mathematical concepts and operations.
It includes a wide range of symbols to represent numbers, operations, relationships, and other critical components like exponents. One common type of mathematical notation, especially in higher mathematics and sciences, is scientific notation.
Scientific notation is particularly useful for handling very large or very small numbers, which frequently appear in scientific and engineering contexts. For example, the number represented as \(6.27 \times 10^{-10}\) uses scientific notation to express a value that would otherwise be quite cumbersome to write out fully.

• This format makes it easier to read, calculate, and communicate data.
• It ensures a clear understanding, improving flexibility in addressing various mathematical problems.

Mastering mathematical notation is fundamental for anyone looking to advance in fields that require a solid grasp of mathematics.

Powers of Ten

Powers of ten are an essential part of scientific notation, simplifying the representation of very large or very small numbers. The notation \(10^n\) signifies a power of ten, where \(n\) is an integer.
The integer \(n\) indicates how many times the number ten is used as a factor. For example, \(10^2\) equals 100, or \(10\times10\), while \(10^{-3}\) corresponds to 0.001, demonstrating repetitive division by ten.

• A positive exponent (e.g., \(10^4\)) magnifies the base of ten, producing large numbers.
• A negative exponent (such as \(10^{-10}\)) diminishes it into a smaller number.

Understanding how powers of ten function is crucial not only for scientific notation but also for broader applications in mathematics, such as calculating scale and proportions efficiently.

Coefficient in Scientific Notation

In scientific notation, the coefficient is a pivotal component that precedes the power of ten, responsible for scaling the base value.
The coefficient should ideally lie between 1 and 10, allowing numbers to be presented in a standardized form that is both precise and concise, such as in the expression \(6.27 \times 10^{-10}\).
Here, \(6.27\) is the coefficient:

• This ensures the expression is straightforward.
• It keeps the number of digits in the coefficient limited, facilitating ease of computation and interpretation.

During the conversion to scientific notation, verifying the coefficient ensures it meets the appropriate standards:

• Check that it falls between 1 and 10.
• Adjust the exponent accordingly if it requires shifting.

Grasping the role and adjustment of the coefficient is integral to mastering scientific notation.