Question

Express these numbers in scientific notation. a) 65 b) -321.09 c) 0.000077099 d) 0.000000000218

Short answer

a) \( 6.5 \times 10^1 \), b) \( -3.2109 \times 10^2 \), c) \( 7.7099 \times 10^{-5} \), d) \( 2.18 \times 10^{-10} \).

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a way to express numbers as a product of a coefficient (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.

Convert 65 to Scientific Notation

For the number 65, which is already greater than 1, we need to convert it so that the coefficient is between 1 and 10. To do this, we move the decimal point one place to the left, giving us 6.5. Therefore, \( 65 = 6.5 \times 10^1 \).

Convert -321.09 to Scientific Notation

To convert -321.09 to scientific notation, we move the decimal point two places to the left to make the coefficient between 1 and 10, resulting in -3.2109. Thus, \( -321.09 = -3.2109 \times 10^2 \).

Convert 0.000077099 to Scientific Notation

For 0.000077099, we move the decimal point five places to the right to obtain a coefficient between 1 and 10, which gives us 7.7099. Therefore, \( 0.000077099 = 7.7099 \times 10^{-5} \).

Convert 0.000000000218 to Scientific Notation

When converting 0.000000000218, we need to move the decimal ten places to the right to get 2.18 as the coefficient. Consequently, \( 0.000000000218 = 2.18 \times 10^{-10} \).

Key concepts

Coefficient

In scientific notation, a number is always expressed as a coefficient multiplied by a power of 10. The coefficient is a crucial part of this format. It is a number that is greater than or equal to 1 but less than 10. This means when you see a number written in scientific notation, like \( 6.5 \times 10^1 \), the \( 6.5 \) is the coefficient. This component defines the significant figures of the number.Some important notes about coefficients:

• It can be positive or negative depending on the original number.
• The coefficient retains the number's precision and magnitude.
• It is always represented in decimal form, ensuring it's within the specified range.
• The position of the decimal point in the coefficient helps define its true value.

Whenever converting a number to scientific notation, it's essential to adjust this coefficient correctly to fit within the range of 1 to 10.

Power of 10

The power of 10 in scientific notation reflects how many times the number should be multiplied or divided by 10 to return to the original value. This component tells the scale or size of the number after adjusting the coefficient.For instance:

• If you move the decimal in a number like 65 to make it a coefficient (6.5), it becomes \( 10^1 \) because you've essentially divided by 10 once.
• Moving the decimal in a small number like 0.000077099 involves multiplying by \( 10^{-5} \), meaning shift the decimal 5 places to the right for normalization.

By understanding the power of 10:

• You can determine whether the original number is very large (positive power) or very small (negative power).
• Each step of change in power represents a tenfold increase or decrease in the number's magnitude.

Thus, correctly calculating the power of 10 is vital in maintaining the accuracy of a number when expressed in scientific notation.

Decimal Point

The decimal point's position is pivotal in converting a number to scientific notation. It defines the coefficient and subsequently, the power of 10.Steps to handle the decimal point:

• Identify where the decimal point needs to be for the coefficient to fall between 1 and 10.
• Shift the decimal left or right to achieve this.
• Keep track of how many places the decimal was moved, which determines the power of 10.

For example:

• In 321.09, the decimal moves two places left to make -3.2109, forming \( -3.2109 \times 10^2 \).
• In 0.000000000218, the decimal moves ten places right to form \( 2.18 \times 10^{-10} \).

It is important to control the decimal point placement precisely, as it ensures the correct representation in scientific notation which in turn preserves the numeric value of the original number.

Integer

In the context of scientific notation, an integer specifically refers to the power of 10 component, denoted as \( n \). This integer represents how many times the decimal point has moved and is essential in signifying the scale of the number.Characteristics of the integer in scientific notation:

• Positive integers indicate a large number, where the decimal was moved left.
• Negative integers indicate a small number, where the decimal was moved right.
• The integer zero means the number is within the range of 1 to 10, needing no shifts.

Each integer affects the outcome radically:

• A number like 65 becomes 6.5 with an integer of \( n=1 \), indicating a shift left and a tenfold decrease.
• For 0.000077099 moving five places means \( n=-5 \), indicating a small number originally.

Recognizing and using the integer correctly ensures that the scientific notation fully represents the original number's size and magnitude.