Question

When a large or small number is written in standard scientific notation, the number is expressed as the product of a number between 1 and \(10,\) multiplied by the appropriate power of \(10 .\) For each of the following numbers, indicate what number between 1 and 10 would be appropriate when expressing the numbers in standard scientific notation. a. 9651 b. 0.003521 c. 93,241 d. 0.000001002

Short answer

The appropriate numbers between 1 and 10 for the given numbers when expressed in standard scientific notation are: a. 9.651 b. 3.521 c. 9.3241 d. 1.002

Step-by-step solution

a. 9651

To express the number 9651 in standard scientific notation, we need to find a number between 1 and 10 and the appropriate power of 10 so that their multiplication results in 9651. We can rewrite 9651 as \(9.651 \times 10^3\). So, the number between 1 and 10 for 9651 is 9.651.

b. 0.003521

To express the number 0.003521 in standard scientific notation, we need to find a number between 1 and 10 and the appropriate power of 10 so that their multiplication results in 0.003521. We can rewrite 0.003521 as \(3.521 \times 10^{-3}\). So, the number between 1 and 10 for 0.003521 is 3.521.

c. 93,241

To express the number 93,241 in standard scientific notation, we need to find a number between 1 and 10 and the appropriate power of 10 so that their multiplication results in 93,241. We can rewrite 93,241 as \(9.3241 \times 10^{4}\). So, the number between 1 and 10 for 93,241 is 9.3241.

d. 0.000001002

To express the number 0.000001002 in standard scientific notation, we need to find a number between 1 and 10 and the appropriate power of 10 so that their multiplication results in 0.000001002. We can rewrite 0.000001002 as \(1.002 \times 10^{-6}\). So, the number between 1 and 10 for 0.000001002 is 1.002.

Key concepts

Standard Scientific Notation

Understanding how to express numbers in standard scientific notation is key in science, engineering, and mathematics. It simplifies the handling of extremely large or small numbers by converting them into a standard form. Standard scientific notation expresses a number as a product of two parts: a coefficient that is a decimal number between 1 and 10, and a power of ten indicated as an exponent.

For example, if we consider the number 9651 from the exercise, it can be written in standard scientific notation as \(9.651 \times 10^3\). The coefficient here is 9.651, and the exponent of ten is 3, which tells us how many times to multiply the coefficient by 10 to get the original number. This method is universally recognized and allows for easier reading and comparison of numbers, regardless of their size.

Powers of Ten

The 'power of ten' is a critical concept when dealing with scientific notation. It refers to the exponent used to denote how many times the number 10 is to be multiplied by itself. A positive power, such as \(10^3\), means we multiply ten three times (10 x 10 x 10), while a negative power, like \(10^{-3}\), denotes division by ten for the given number of times (1 / (10 x 10 x 10)).

In our exercise, the power of ten changes with each number to keep the coefficient between 1 and 10. For instance, the small number 0.003521 becomes \(3.521 \times 10^{-3}\), demonstrating the inverse relationship for negative exponents when expressing small numbers.

Expressing Large Numbers

When it comes to large numbers, scientific notation is an efficient way to express and manage them. Writing out zeroes for very large quantities can be cumbersome and prone to error. With scientific notation, you abbreviate the number using a power of ten.

Consider the number 93,241 from the exercise, this can be written as \(9.3241 \times 10^{4}\), reducing the lengthy numeral to a concise form. This makes mathematical operations with large values more practical, whether you're calculating the distance between stars or processing financial data.

Expressing Small Numbers

Similar to large numbers, very small numbers can also be difficult to read and work with. Scientific notation provides a solution by allowing us to express these figures in a compact form. In scientific notation, small numbers are shown with a negative exponent.

For example, the tiny number 0.000001002 from the exercise is represented as \(1.002 \times 10^{-6}\). This illustrates the convenience of scientific notation, transforming a decimal with many leading zeroes into a simple expression, making it easier to compare, and work with, tiny quantities such as the size of molecules or electrical charges in physics.