Question

Convert the following values to scientific notation: (a) \(0.058 \mathrm{~g}\) (b) \(46,792 \mathrm{~m}\) (c) \(0.006072 \mathrm{~cm}\) (d) \(345.3 \mathrm{~kg}\)

Short answer

(a) \(5.8 \times 10^{-2}\), (b) \(4.6792 \times 10^4\), (c) \(6.072 \times 10^{-3}\), (d) \(3.453 \times 10^2\).

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a way to express numbers that are too large or too small conveniently. For any number, this involves rewriting it as a number between 1 and 10 multiplied by a power of 10. The process involves shifting the decimal point until only one non-zero digit remains on its left.

Converting 0.058 g

To express 0.058 in scientific notation, shift the decimal point to the right until the number becomes a number between 1 and 10. This means moving it two places to the right:\[0.058 = 5.8 \times 10^{-2}\]The exponent is negative because the original number is less than 1.

Converting 46,792 m

For 46,792, move the decimal point 4 places to the left to turn it into a number between 1 and 10:\[46,792 = 4.6792 \times 10^4\]The exponent is positive because the original number is greater than 1, and you moved the decimal point left.

Converting 0.006072 cm

To convert 0.006072 to scientific notation, move the decimal point 3 places to the right:\[0.006072 = 6.072 \times 10^{-3}\]The exponent is negative because the number was initially less than 1.

Converting 345.3 kg

For 345.3, shift the decimal point 2 places to the left:\[345.3 = 3.453 \times 10^2\]The exponent is positive because the number was initially greater than 1.

Key concepts

Decimal Point

In scientific notation, the decimal point plays a crucial role by determining the position of the number in terms of powers of ten. When converting a number to scientific notation, shifting the decimal point is essential for correctly expressing the number.

• If you move the decimal point to the right, it indicates that the original number is smaller than 1.
• If you move it to the left, it signals that the number is larger than 1.

Whenever you shift the decimal point, the resulting number must fall between 1 and 10 for proper scientific notation form. This ensures clarity and standardizes the representation of both large and small numbers.

Exponent

The exponent in scientific notation expresses how many times the power of ten needs to be multiplied by the base to obtain the original number. It is written as a superscript following the base number.

• If you move the decimal point to the left, the exponent is positive.
• If you move it to the right, the exponent is negative.

For example, when converting 345.3 to scientific notation, the process involves moving the decimal two places to the left, resulting in an exponent of 2. Thus, it becomes \(3.453 \times 10^2\). This highlights how the exponent captures the magnitude of the number's scale factor.

Powers of Ten

The powers of ten are the backbone of scientific notation, serving as the scale factor that adjusts the size of the base number. Each shift in the decimal point corresponds to a change in the power of ten.
When a number is multiplied by \(10^n\), where \(n\) is the exponent:

• A positive \(n\) indicates that the decimal point shifted to the left.
• A negative \(n\) suggests that the decimal point shifted to the right.

For instance, the number 46,792 can be expressed as \(4.6792 \times 10^4\), demonstrating a four-place shift in the decimal to the left, adjusting the number's scale by a factor of 10,000.

Large Numbers

When dealing with large numbers, scientific notation simplifies their representation, making them easier to read and work with, especially in calculations. Large numbers are those greater than 1 and involve a positive exponent.
For example, in converting 46,792 to scientific notation, the decimal point moves four places to the left, creating a number between 1 and 10 with a corresponding exponent of 4. This conversion results in \(4.6792 \times 10^4\).

• This process reduces the complexity of working with such large values.
• It also facilitates comparisons, as it standardizes the way numbers are presented.

Small Numbers

In contrast to large numbers, small numbers in scientific notation have a value less than 1 and are represented with a negative exponent. This format helps manage very small values effectively.
When converting these numbers, you shift the decimal point to the right, allowing for the creation of a number between 1 and 10. For instance, the number 0.058 transforms into \(5.8 \times 10^{-2}\) as the decimal point shifts two places to the right.

• This helps in expressing very small quantities in a standardized form.
• It enhances the ease of performing mathematical operations on tiny measurements.