Problem 14 - Algebra 1

Question

Rewrite in scientific notation. $$ 0.000000006 $$

Step-by-Step Solution

Answer

The number \(0.000000006\) in scientific notation is \(6 \times 10^{-9}\).

Step 1: Identify the Coefficient and Decimal Movement

Determine the coefficient by establishing the first number and the number of places to move the decimal. The first number should be between 1 and 10, so the coefficient for \(0.000000006\) will be \(6\). To move the decimal point from after the 6 to its new position after the leading digit, it takes 9 moves. Hence, \(0.000000006\) will be \(6 \times 10^{-9}\).

Step 2: Write in Scientific Notation

Use the coefficient and decimal places to compose the number into scientific notation. Since there are nine places to move the decimal and it is moved to the right, the exponent of 10 will be -9. Therefore, the number in scientific notation is \(6 \times 10^{-9}\).

Key Concepts

Decimal Movement

When converting a number into scientific notation, understanding decimal movement is crucial. This refers to how many places the decimal point has to be shifted to give the number a coefficient between 1 and 10. Let's take the example of the number 0.000000006.
To convert this number into scientific notation, we need to shift the decimal point such that only one non-zero digit appears to the left of the decimal. In this case, the number is 6, which is achieved by moving the decimal 9 places to the right.

• Count the number of positions the decimal point needs to move.
• It should be moved until a number between 1 and 10 remains as the coefficient.
• In 0.000000006, the decimal moves 9 places to get 6.0000000.

The movement direction (left or right) is important as it indicates whether the number is smaller or larger than one, thereby affecting the exponent sign.

Coefficient Identification

Coefficient identification is an essential step in writing numbers in scientific notation. The idea is to represent the number as a product of a coefficient and a power of 10. The coefficient must be a number between 1 and 10.
In mathematical terms, a coefficient is simply the number that multiplies the power of ten. For the number 0.000000006, once we move the decimal point 9 places to the right, we get the coefficient 6.

• Ensure the chosen coefficient is between 1 and 10.
• This coefficient is derived after adjusting the decimal point appropriately.
• For our example, 6 is the number found by moving the decimal point.

Identifying the coefficient ensures accuracy in representing the number's size in scientific notation.

Negative Exponent

Scientific notation uses exponents to indicate how many places the decimal has moved. A negative exponent is used when the original number is less than one. This signifies that the decimal was shifted to the right.
For the number 0.000000006, the decimal moved 9 places to the right, thus the exponent becomes -9.

• Negative exponents reflect decimal movement to the right.
• They are indicators that the number is very small.
• Here, the scientific notation ends as \(6 \times 10^{-9}\).

Negative exponents play a key role in expressing very small numbers efficiently and succinctly in scientific notation.