Question

Express these numbers in scientific notation: (a) 0.000000027 , (b) 356 (c) 0.096 .

Short answer

(a) \(2.7 \times 10^{-8}\) (b) \(3.56 \times 10^2\) (c) \(9.6 \times 10^{-1}\)

Step-by-step solution

Expression of 0.000000027 in Scientific Notation

Move the decimal 8 places to the right to get 2.7, which gives \(2.7 \times 10^{-8}\).

Expression of 356 in Scientific Notation

Move the decimal 2 places to the left to get 3.56, which gives \(3.56 \times 10^2\).

Expression of 0.096 in Scientific Notation

Move the decimal 1 place to the right to get 9.6, which gives \(9.6 \times 10^{-1}\).

Key concepts

Decimal Places

When we talk about decimal places, we're referring to the position of a number after the decimal point. These positions help to indicate the precision of a number. In scientific notation, understanding decimal places is crucial because you'll often change the position of the decimal as part of writing the number in this format.

For example:

• In the number 0.000000027, the decimal moves 8 places to the right to reach the first non-zero digit, producing 2.7.
• In 0.096, it moves 1 place to the right to get 9.6.
• For a whole number like 356, you move the decimal 2 places to the left to convert it into 3.56.

Moving the decimal correctly determines the exponent in scientific notation, which we'll explore next.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a power, often represented with a base and an exponent. This is crucial in scientific notation.

In scientific notation, the exponent indicates how many times the base (10) must be multiplied by itself. For instance:

• The notation for 0.000000027 as 2.7 involves moving the decimal point 8 places, resulting in the exponent \(-8\) in scientific notation, giving us \(2.7 \times 10^{-8}\).
• For 0.096, moving the decimal one place gives an exponent of \(-1\), resulting in \(9.6 \times 10^{-1}\).
• With the number 356, the decimal moves two places left, resulting in \(3.56 \times 10^2\).

Understanding how to determine the correct exponent is essential for accurately expressing numbers in scientific notation.

Standard Form

Standard form is another term for scientific notation, widely used in mathematics and sciences to simplify the representation of very large or very small numbers.

In standard form, numbers are written as a product of a number (between 1 and 10) and a power of 10. You'll often convert between the two forms for clarity and simplicity.

• For example, the small number 0.000000027 is hard to interpret at a glance, but in standard form, it becomes \(2.7 \times 10^{-8}\), making it clear how small it truly is.
• A number like 0.096 becomes \(9.6 \times 10^{-1}\) in standard form, again simplifying its representation.
• Conversely, a larger number such as 356 can be easily managed as \(3.56 \times 10^2\), showing its relative size in a more understandable way.

The use of standard form is particularly helpful when dealing with calculations involving several zeros, allowing for easier reading and handling of numbers. Knowing how to express numbers in standard form is an essential skill in numerous scientific and engineering fields.