Question

Express the following numbers in scientific notation: (a) 0.000000027 , (b) 356 , (c) 47,764, (d) 0.096 .

Short answer

a) \( 2.7 \times 10^{-8} \), b) \( 3.56 \times 10^2 \), c) \( 4.7764 \times 10^4 \), d) \( 9.6 \times 10^{-2} \).

Step-by-step solution

Understanding Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10. The general form is \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.

Converting 0.000000027 to Scientific Notation

Move the decimal point in 0.000000027 to the right until you have a number between 1 and 10. Count the number of places you moved the decimal point, which is 8 places. The scientific notation is \( 2.7 \times 10^{-8} \).

Converting 356 to Scientific Notation

Move the decimal point in 356 to the left until it is between 1 and 10. Since the decimal must be after the first '3', the number becomes 3.56 and the decimal point moves 2 places. Thus, \( 356 \) is \( 3.56 \times 10^2 \) in scientific notation.

Converting 47,764 to Scientific Notation

Move the decimal point in 47,764 to the left until it is after the 4, giving the number 4.7764. Since the decimal point moves 4 places to the left, the scientific notation is \( 4.7764 \times 10^4 \).

Converting 0.096 to Scientific Notation

Move the decimal point in 0.096 to the right until the number is between 1 and 10, resulting in 9.6. This requires moving the decimal 2 places, so the scientific notation is \( 9.6 \times 10^{-2} \).

Key concepts

Decimal Place Movement

When converting a number to scientific notation, a crucial step involves moving the decimal point. This movement helps transform the number to be between 1 and 10, which is required for scientific notation. For example, take the number 0.000000027. To convert this into scientific notation, you need to move the decimal point 8 places to the right, resulting in the number 2.7. This is because you want to find a number between 1 and 10.
Similarly, for a number like 356, you move the decimal point 2 places to the left to make it 3.56. Each shift of the decimal point corresponds to a place value adjustment expressed as a power of ten. Keep in mind:

• Moving the decimal to the right results in a negative power of ten.
• Moving the decimal to the left results in a positive power of ten.

This process of decimal adjustment is vital for expressing very small or very large numbers in a compact, convenient form using scientific notation.

Powers of Ten

The crux of scientific notation is its reliance on powers of ten to represent numerical values succinctly. The power of ten shows how many times the base number, 10, is used as a factor. For instance, when you shift the decimal point 8 places to the right for the number 0.000000027, you express this as a power of ten of \( 10^{-8} \).
This indicates that you're dealing with a very small number. On the other hand, for a number like 47,764, moving the decimal point 4 places to the left means you use \( 10^{4} \). This expresses a large number succinctly. Here are some important points:

• A positive exponent suggests a larger number.
• A negative exponent implies a smaller number.

By coupling a number with an appropriate power of ten, scientific notation allows for easy communication and calculation with numbers that are excessively large or small.

Number Format Conversion

Number format conversion involves changing a number from its standard decimal form to that of scientific notation, which consists of two components: a coefficient and a power of ten. This conversion is essential in fields such as science and engineering where dealing with extremes in data size is common.
To convert a number like 0.096 to scientific notation, you adjust the number to be between 1 and 10, resulting in 9.6. You then express the movement of the decimal point as a power of ten, \( 10^{-2} \). Thus, 0.096 converts to \( 9.6 \times 10^{-2} \).
The steps for converting numbers are generally as follows:

• Move the decimal point to achieve a coefficient (a number) between 1 and 10.
• Count how many places you moved the decimal point – this gives the power of ten.
• Use positive powers for large numbers and negative powers for small numbers.

Mastering number format conversion makes it easier to handle calculations across varied magnitudes and simplifies otherwise complex arithmetic operations.