Question

Write each of the following numbers in standard scientific notation. a. \(4381 \times 10^{-4}\) b. \(98,784 \times 10^{4}\) c. \(78.21 \times 10^{2}\) d. \(9.871 \times 10^{-4}\) e. \(0.009871 \times 10^{7}\) f. \(42,221 \times 10^{4}\) g. \(0.00008951 \times 10^{6}\) h. \(0.00008951 \times 10^{-6}\)

Short answer

a. \(4.381 \times 10^{-1}\) b. \(9.8784 \times 10^{8}\) c. \(7.821 \times 10^{3}\) d. \(9.871 \times 10^{-4}\) e. \(9.871 \times 10^{4}\) f. \(4.2221 \times 10^{8}\) g. \(8.951 \times 10^{1}\) h. \(8.951 \times 10^{-11}\)

Step-by-step solution

Adjust the decimal point

Move the decimal point so that the given number becomes a number between 1 and 10.

Multiply the adjusted number with the power of 10

Multiply the adjusted number by the given power of 10 in the exercise.

Find the new exponent

Determine the number of decimal places the decimal point was moved and add or subtract that value from the given exponent to find the new exponent. Now, let's use these steps to solve each part of the exercise. a. \(4381 \times 10^{-4}\) b. \(98,784 \times 10^{4}\) c. \(78.21 \times 10^{2}\) d. \(9.871 \times 10^{-4}\) e. \(0.009871 \times 10^{7}\) f. \(42,221 \times 10^{4}\) g. \(0.00008951 \times 10^{6}\) h. \(0.00008951 \times 10^{-6}\)

Solution for a

1. Move 3 decimal places to the left: 4.381 2. Multiply by \(10^{-4}\): \(4.381 \times 10^{-4}\) 3. Add 3 to the exponent \(-4\): \(4.381 \times 10^{-1}\)

Solution for b

1. Move 4 decimal places to the left: 9.8784 2. Multiply by \(10^{4}\): \(9.8784 \times 10^{4}\) 3. Add 4 to the exponent 4: \(9.8784 \times 10^{8}\)

Solution for c

1. Move 1 decimal place to the left: 7.821 2. Multiply by \(10^{2}\): \(7.821 \times 10^{2}\) 3. Add 1 to the exponent 2: \(7.821 \times 10^{3}\)

Solution for d

1. No adjustments needed: 9.871 2. Multiply by \(10^{-4}\): \(9.871 \times 10^{-4}\) 3. No change to the exponent: \(9.871 \times 10^{-4}\)

Solution for e

1. Move 3 decimal places to the right: 9.871 2. Multiply by \(10^{7}\): \(9.871 \times 10^{7}\) 3. Subtract 3 from the exponent 7: \(9.871 \times 10^{4}\)

Solution for f

1. Move 4 decimal places to the left: 4.2221 2. Multiply by \(10^{4}\): \(4.2221 \times 10^{4}\) 3. Add 4 to the exponent 4: \(4.2221 \times 10^{8}\)

Solution for g

1. Move 5 decimal places to the right: 8.951 2. Multiply by \(10^{6}\): \(8.951 \times 10^{6}\) 3. Subtract 5 from the exponent 6: \(8.951 \times 10^{1}\)

Solution for h

1. Move 5 decimal places to the right: 8.951 2. Multiply by \(10^{-6}\): \(8.951 \times 10^{-6}\) 3. Subtract 5 from the exponent \(-6\): \(8.951 \times 10^{-11}\) Final answers in standard scientific notation: a. \(4.381 \times 10^{-1}\) b. \(9.8784 \times 10^{8}\) c. \(7.821 \times 10^{3}\) d. \(9.871 \times 10^{-4}\) e. \(9.871 \times 10^{4}\) f. \(4.2221 \times 10^{8}\) g. \(8.951 \times 10^{1}\) h. \(8.951 \times 10^{-11}\)

Key concepts

Standard Scientific Notation

Standard scientific notation is a concise way to express very large or very small numbers, frequently used in science and engineering. It's also a fantastic tool for easily comparing the magnitude of different values. The format for this notation includes a single nonzero digit to the left of the decimal point, followed by any additional significant digits. After this, the number is multiplied by 10 raised to an exponent. This exponent denotes how many places the decimal point needs to be shifted to revert the number to its original value.

For example, the number 450,000 becomes 4.5 multiplied by 10 to the power of 5, written as \(4.5 \times 10^5\). Conversely, a small number like 0.00032 is written as 3.2 multiplied by 10 to the power of -4, or \(3.2 \times 10^{-4}\).

Exponents in Scientific Notation

In scientific notation, the exponent part is crucial as it tells us the magnitude of the number. When a number is written in scientific notation, the exponent on the 10 indicates how many places to move the decimal point to return to the original value. A positive exponent, such as \(10^3\), suggests moving the decimal point to the right, making the number larger. Meanwhile, a negative exponent, such as \(10^{-3}\), tells us to move the decimal to the left, resulting in a smaller number.

Example:

Let's take \(3.6 \times 10^2\), where the exponent is +2. We move the decimal in 3.6 two places to the right to get the original number, 360. Conversely, for \(2.1 \times 10^{-3}\), we move the decimal in 2.1 three places to the left, yielding 0.0021.

Decimal Point Adjustment

Adjusting the decimal point is a key step in writing a number in standard scientific notation. To do this, you must decide the new position of the decimal point so that there's only one non-zero digit in front of it. After moving the decimal to achieve this format, count how many places you moved it. This count then determines the exponent for 10 in your final expression.

For instance, when converting 7,300 to standard scientific notation, you'll move the decimal point three places to the left to get 7.3. Since you moved the decimal to the left, your exponent will be positive. The scientific notation is \(7.3 \times 10^3\). Similarly, to convert 0.0042, you'll move the decimal three places to the right to get 4.2. Here, the decimal moved to the right, so the exponent will be negative, giving you \(4.2 \times 10^{-3}\).

Understanding these concepts of scientific notation ensures not just correct conversions but also provides a solid foundation for deeper mathematical and scientific learning.