Question

Write each of the following in scientific notation: a. \(55000 \mathrm{~m}\) b. \(480 \mathrm{~g}\) c. \(0.000005 \mathrm{~cm}\) d. \(0.00014 \mathrm{~s}\) e. \(0.0072 \mathrm{~L}\) f. \(670000 \mathrm{~kg}\)

Short answer

a) 5.5 × 10^4 mb) 4.8 × 10^2 gc) 5 × 10^{-6} cmd) 1.4 × 10^{-4} se) 7.2 × 10^{-3} Lf) 6.7 × 10^5 kg

Step-by-step solution

Understanding Scientific Notation

Scientific notation expresses numbers as a product of a coefficient (between 1 and 10) and a power of ten. For example, 55000 can be written as 5.5 × 10^4 in scientific notation.

Converting 55000 meters

Move the decimal point in 55000 to the left until one digit remains on its left. This gives 5.5. Count the number of places moved, which is 4. Thus, 55000 meters is written as 5.5 × 10^4 meters in scientific notation.

Converting 480 grams

Move the decimal point to the left two places until one digit remains on its left, resulting in 4.8. So, 480 grams is written as 4.8 × 10^2 grams in scientific notation.

Converting 0.000005 centimeters

Move the decimal point to the right six places until one digit remains on its left, resulting in 5. The number of places moved is -6. Thus, 0.000005 centimeters is written as 5 × 10^{-6} centimeters.

Converting 0.00014 seconds

Move the decimal point to the right four places until one digit remains on its left, resulting in 1.4. The number of places moved is -4. Hence, 0.00014 seconds is written as 1.4 × 10^{-4} seconds.

Converting 0.0072 liters

Shift the decimal point to the right three places until one digit remains on its left, resulting in 7.2. The number of places moved is -3. Therefore, 0.0072 liters is written as 7.2 × 10^{-3} liters.

Converting 670000 kilograms

Move the decimal point to the left five places until one digit remains on its left, resulting in 6.7. So, 670000 kilograms is written as 6.7 × 10^5 kilograms.

Key concepts

decimal point

The decimal point is a critical component when working with scientific notation. It's a period or dot used to separate the integer part of a number from its fractional part. In scientific notation, we move the decimal point to form a number between 1 and 10. This new number is known as the coefficient.

For example, in converting 55000 meters to scientific notation, we move the decimal point four places to the left to change the number to 5.5. This movement is necessary because in scientific notation, the coefficient must always be a number between 1 and 10.

Understanding how to manipulate the decimal point will help you easily convert numbers to and from scientific notation. Remember to count how many places you’ve moved the decimal point. This count will be the exponent for the power of ten in the scientific notation format.

coefficient

In scientific notation, the coefficient is the number that results after moving the decimal point. It is always between 1 and 10, inclusive.

Let's look at the number 0.000005 centimeters. To convert it to scientific notation, we move the decimal point six places to the right until we get the number 5. This makes 5 the coefficient in the scientific notation form, which would be written as 5 × 10^{-6} centimeters.

The coefficient ensures that the number remains manageable and easy to work with, especially when dealing with very large or very small numbers. By consistently using a coefficient between 1 and 10, we maintain a standardized format that is universally understood in scientific communities.

power of ten

The power of ten represents how many places we moved the decimal point to convert a number into scientific notation. This exponent indicates the magnitude or scale of the original number.

For instance, in the conversion of 480 grams to scientific notation, we move the decimal point two places to the left to get 4.8. Since we moved it two places, the power of ten will be 10^2. Hence, 480 grams is written as 4.8 × 10^2 grams.

Similarly, if we move the decimal point to the right, the power of ten becomes negative. When converting 0.00014 seconds, we move the decimal four places to the right, which makes the power of ten -4. Thus, 0.00014 seconds becomes 1.4 × 10^{-4} seconds.

Understanding the power of ten is crucial for accurately representing the original number’s scale in scientific notation. It conveys whether the original number was large or small and by what magnitude.