Problem 40 - Elementary and Intermediate Algebra

Question

Write number in scientific notation. 0.0001

Step-by-Step Solution

Answer

1.0 × 10^{-4}

Step 1: Identify the Number

Here, the number we need to write in scientific notation is 0.0001.

Step 2: Determine the Place Value

We need to determine the place value that makes the number between 1 and 10. In this case, moving the decimal point 4 places to the right gives us 1.0.

Step 3: Express the Number as a Product of a Decimal and a Power of 10

Once you have the decimal (1.0), express the number as a product of this decimal and a power of ten. Since we moved the decimal point 4 places to the right, the power of ten will be 1.0 imes 10^{-4}

Step 4: Write the Final Scientific Notation

Combine the results from all the steps to express the number in scientific notation. The scientific notation for 0.0001 is 1 imes 10^{-4}.

Key Concepts

Decimal Point Movement

When you write a number in scientific notation, the first step involves moving the decimal point. The goal here is to turn your number into a decimal that is between 1 and 10.

For example, if you start with the number 0.0001, you need to move the decimal point so that it lands right after the first non-zero digit. In this case, you move the decimal four places to the right, transforming the number into 1.0. This step is crucial because it sets you up to express the number in terms of a power of ten.

Always remember:

• Move the decimal to the right, if the number is smaller than 1.
• Move the decimal to the left, if the number is larger than 10.

Knowing how to move the decimal point correctly is a key skill in mastering scientific notation.

Power of Ten

The power of ten is a fundamental aspect of scientific notation. It represents how many places you've moved the decimal point. If you moved the decimal to the right, like with 0.0001, the power of ten will be negative.

Here's why: each move to the right diminishes the original value of the number by a factor of ten.

For 0.0001:

• The decimal moved 4 places to the right, which means the exponent for the power of ten is -4.

So, in the scientific notation, it's written as \[1.0 \times 10^{-4}\].

This tells you exactly how many times to divide the number by ten to return to the original value. It's a handy way to understand very small or large quantities quickly.

Place Value

Understanding place value in numbers is important for scientific notation. When you change the place of a decimal point, you're essentially changing the place value of the digits in the number.

Consider the number 0.0001:

• Originally, the 1 is in the ten-thousandths place.
• After shifting the decimal point to create 1.0, the 1 is now in the units place.

This change in place value is what allows you to represent the number neatly as a power of ten, making it easier to read and comprehend when dealing with extremely large or small figures. By adjusting place value, scientific notation simplifies numbers without losing their meaning.

Always keep in mind:

• Smaller numbers (less than 1) will have negative powers of ten.
• Larger numbers (greater than 10) will have positive powers of ten.

Mastering place value transformation helps in seamlessly dealing with scientific notation.