Problem 9 - Algebra 1: Concepts and Skills

Question

Write the number in scientific notation. $$ 6.900 .000 $$

Step-by-Step Solution

Answer

The number 6900000 in scientific notation is \(6.9 \times 10^6\).

Step 1: Identify the decimal point

Identify the location of the decimal point in your original number. In this case, the number is 6900000 so the decimal point is at the end of this number.

Step 2: Move the decimal point

Move the decimal point in your original number to the right of the first non-zero digit. Every time the point is moved, count the places. Keep moving until you have a number between 1 and 10. With the number 6900000, moving the decimal point 6 places to the left gives 6.9

Step 3: Convert to scientific notation

Write the number obtained from step 2, then multiply it by \(10\) raised to the power equal to the number of moves made in step 2. This means 6.9 becomes \(6.9 \times 10^6\).

Key Concepts

Decimal Point

Understanding the decimal point is key when working with numbers in scientific notation. The decimal point separates the whole number part from the fractional part, but sometimes the point is hidden or needs to be repositioned.
In the number 6900000, the decimal point is initially at the end, following the zeros. To convert the number into scientific notation, you'll need to move this decimal point. The goal is to end up with a number between 1 and 10.

Whenever you move the decimal point, count the shifts you make. This will be important later when you determine the exponent.

• The decimal point in 6900000 moves left until 6.9 is formed.
• This process simplifies big numbers, making them easier to work with.

Mastery of the decimal point's movement is essential for number conversion tasks.

Exponents

Exponents are a compact way to express repeated multiplication. In scientific notation, exponents of 10 are used to show how many times the decimal point is moved.
For the number 6900000, moving the decimal point six places to the left means the exponent will be 6. This is represented as \(10^6\).

Exponents make it easier to handle extremely large or small numbers, by transforming them into manageable forms.

• When moving the decimal point left, the exponent is positive.
• When moving the decimal right, the exponent is negative.

Remember, the exponent determines the amount of zeros in the number when reverting to standard form.

Number Conversion

Number conversion involves translating a number from one form to another, such as from standard to scientific notation. This notation is helpful for expressing both very large and very small numbers conveniently.
The steps to convert:

• Identify the decimal point's current location.
• Move it to position the number between 1 and 10.
• Count the shifts and use this count as the exponent of 10.

For 6900000, after moving the decimal point from the end to just after the 6, the number becomes 6.9. The exponent tells us how many places the point was moved, which is 6, so it transforms to \(6.9 \times 10^6\).
This streamlined method keeps your calculations simple while allowing for flexibility when working with diverse numbers.