Question

For each of the following numbers, by how many places must the decimal point be moved to express the number in standard scientific notation? In each case, will the exponent be positive, negative, or zero? a. 55,651 b. 0.000008991 c. 2.04 d. 883,541 e. 0.09814

Short answer

a. Move decimal point 4 places, exponent = +4: \(5.5651 \times 10^4\) b. Move decimal point 6 places, exponent = -6: \(8.991 \times 10^{-6}\) c. No movement, exponent = 0: \(2.04 \times 10^0\) d. Move decimal point 5 places, exponent = +5: \(8.83541 \times 10^5\) e. Move decimal point 2 places, exponent = -2: \(9.814 \times 10^{-2}\)

Step-by-step solution

a. 55,651

To express 55,651 in standard scientific notation, follow these steps: 1. Move the decimal point between the first two non-zero digits (from right to left): 5.5651 2. Count the number of places moved: 4 3. The exponent will be positive because the original number is greater than 1: \(5.5651 \times 10^4\)

b. 0.000008991

To express 0.000008991 in standard scientific notation, follow these steps: 1. Move the decimal point between the first two non-zero digits (from left to right): 8.991 2. Count the number of places moved: 6 3. The exponent will be negative because the original number is less than 1: \(8.991 \times 10^{-6}\)

c. 2.04

To express 2.04 in standard scientific notation, follow these steps: 1. The decimal point is already between the first two non-zero digits: 2.04 2. The number of places moved: 0 3. The exponent will be zero because no movement of the decimal point took place: \(2.04 \times 10^0\)

d. 883,541

To express 883,541 in standard scientific notation, follow these steps: 1. Move the decimal point between the first two non-zero digits (from right to left): 8.83541 2. Count the number of places moved: 5 3. The exponent will be positive because the original number is greater than 1: \(8.83541 \times 10^5\)

e. 0.09814

To express 0.09814 in standard scientific notation, follow these steps: 1. Move the decimal point between the first two non-zero digits (from left to right): 9.814 2. Count the number of places moved: 2 3. The exponent will be negative because the original number is less than 1: \(9.814 \times 10^{-2}\)

Key concepts

Understanding the Decimal Point in Scientific Notation

The decimal point plays a crucial role in expressing numbers in scientific notation. In fact, it controls where we position the number so that only one non-zero digit stands before it. Understanding where to place this decimal point is key.

• First, identify the first non-zero digit in your number.
• Move the decimal point right behind this digit.

By doing this, you are effectively simplifying the number so it can be easily notated with an exponent of ten.
For instance, take the number 55,651. The decimal is currently at the end, so we move it four places to the right of 5 to get 5.5651. The number of places moved will determine your exponent.

Exploring the Role of Exponents

In scientific notation, exponents tell us how many places and in which direction the decimal point was moved. This small superscript is powerful as it makes large numbers manageable and small numbers comprehensible.

• A positive exponent indicates that the decimal was moved to the left for larger numbers.
• A negative exponent indicates that the decimal was moved to the right for smaller numbers.
• A zero exponent means the decimal point did not move at all.

For example, when expressing 0.000008991, the decimal moves six places left to 8.991, giving us a negative exponent of -6. Thus, it is expressed as \(8.991 \times 10^{-6}\), which simplifies it greatly.

The Importance of Non-Zero Digits

Non-zero digits in scientific notation serve as anchors. They help to clarify significant figures which represent the precision of a number.

• They ensure accuracy in representing the number's value.
• Only one non-zero digit is placed before the decimal point in standard scientific notation.

Consider the example of 0.09814. The first non-zero digit is 9. By moving the decimal point two places right to sit just behind 9, it reads as 9.814. This movement results in an exponent of -2 as the number is originally less than 1. Thus, the standard form is \(9.814 \times 10^{-2}\). Non-zero digits give your number its value and differentiate it from others.