Problem 39 - Elementary and Intermediate Algebra

Question

Write number in scientific notation. 0.0000003

Step-by-Step Solution

Answer

0.0000003 is written as \(3 \times 10^{-7}\) in scientific notation.

Step 1: Identify the base number

First, identify the significant digits in the number 0.0000003, which is 3 because it is the first non-zero digit.

Step 2: Determine the power of 10

Count how many decimal places you would need to move the decimal point to place it just after the first significant digit. For 0.0000003, you move the decimal 7 places to the right to get the base number 3.

Step 3: Write in scientific notation

Combine the base number with the power of ten that reflects how many places the decimal moved. Since the decimal moved 7 places to the right, the power of 10 will be negative: \[3 \times 10^{-7}\].

Key Concepts

Understanding Significant Digits

In the context of scientific notation, significant digits play a crucial role. They represent the meaningful figures in a number, all non-zero digits in a number are significant. When you look at the number 0.0000003, the significant digit is '3' because it is the first and only non-zero digit.

Significant digits tell us about both the precision and accuracy of a number or measurement. Here are some basic rules:

• All non-zero digits are significant.
• Zeros between non-zero digits are significant.
• Leading zeros, which precede non-zero digits, are not significant.
• Trailing zeros in a number with a decimal point are significant.

In scientific notation, you want to keep all the significant digits from the original number because they convey essential information about its precision.

Exploring the Power of Ten

The power of ten is a fundamental concept when writing numbers in scientific notation. It describes how many times you need to multiply the number by 10 to shift the decimal point, aiding in managing large or very small numbers.

For example, in a number like 0.0000003, you need to shift the decimal point 7 places to the right to set it after the significant digit '3'. Therefore, the power of ten becomes -7.

Here's a little guide:

• When you move the decimal to the right, the power of ten is negative.
• When you move the decimal to the left, the power of ten is positive.

This helps clearly express very small (or very large) numbers in a much more manageable form, where the input and magnitude can be easily interpreted.

Reading Decimal Places

Decimal places are important when converting numbers to scientific notation, as they dictate where the decimal point should be placed in the scientific form.

Consider the number 0.0000003. To convert it to scientific notation, we place the decimal right after the first significant digit, '3'. This requires moving it 7 places—counting every zero that appears after the decimal and before the '3'—resulting in a negative power of ten.

This process is straightforward once you understand it:

• If there are decimal places before the significant number, like in 0.00003, count each place.
• The number of movements equals the exponent in the power of ten.

Mastering this can make interpreting and manipulating scientific data much easier and is a vital skill for dealing with both large and small measurements efficiently.