Problem 16 - Algebra for College Students

Question

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(0.00000082\)

Step-by-Step Solution

Answer

The scientific notation of 0.00000082 is \(8.2 \times 10^{-7}\).

Step 1: Identify the non-zero digits

In the number 0.00000082, locate the non-zero digits. The non-zero digits are 8 and 2.

Step 2: Place the non-zero digits after the decimal point

Rearrange the identified non-zero digits as a single number that will fall between 1 and 10. This becomes 8.2.

Step 3: Count the decimal places

Count how many decimal places you move the decimal point to get from 0.00000082 to 8.2. Here, the decimal is moved 7 places to the right.

Step 4: Write in scientific notation

Express the number as a product of the number from Step 2 and 10 raised to the power of the negative number determined in Step 3. This yields the expression: \(8.2 \times 10^{-7}\).

Key Concepts

Understanding Decimal Places

Decimal places are the number of digits to the right of a decimal point in a number. They help us understand the value of the number more precisely.
In scientific notation, identifying how many times we move the decimal point is crucial as it affects the exponent of ten.
For the number 0.00000082, the decimal point moves seven places to the right to transform the number into 8.2 for scientific notation.

• We initially have "0." followed by six zeros, placing the original value quite far from 1.
• Each move of the decimal represents a power of ten change.

This shift is translated into negative exponents, as the original number is less than one. Thus, for 0.00000082, the decimal moves 7 places right, resulting in the exponent -7.

Importance of Non-zero Digits

Non-zero digits are crucial because they hold the significant value of a number.
In a number like 0.00000082, the only non-zero digits are 8 and 2.
To express a number in scientific notation, these digits must form a number between 1 and 10.

• We ignore leading zeros as they do not affect the measurement of significant figures.
• By placing non-zero digits first, we ensure the base number is appropriately scaled between 1 and 10.

Thus, 8.2 becomes our base number in scientific notation, preserving the magnitude of the original digits.

Making Sense of Exponents

Exponents in scientific notation describe how many times the number 10 is used as a factor.
They allow us to express very large or very small numbers concisely and make calculations easier.
For the number 0.00000082, moving the decimal point seven places means we use the power of 10 raised to -7.

• Negative exponents indicate numbers less than one. Each decrement of the exponent represents moving the decimal left.
• This is opposite to positive exponents, which imply moving the decimal right, used for numbers greater than one.

Therefore, 0.00000082 is written as \(8.2 \times 10^{-7}\), capturing both the magnitude and direction shift of the decimal.