Problem 83
Question
Write the number in scientific notation. (Lesson 8.5) $$ 0.0000008 $$
Step-by-Step Solution
Verified Answer
The number 0.0000008 in scientific notation is \(8 \times 10^{-7}\).
1Step 1: Identify the Decimal Point and Direction of Movement
Identify the position of the decimal point in the number 0.0000008. It is located after the 0 and before the eight zeros. The decimal needs to be moved 7 places to the right to situate it after the 8.
2Step 2: Count the Number of Movements
As the direction is to the right, this indicates that the exponent of 10 will be negative (as this denotes smaller quantities in scientific notation). Count the number of places move. This total is 7, so the exponent of 10 is -7.
3Step 3: Write the Number in Scientific Notation
Now, write down the number in scientific notation. After moving 7 places becomes 8. The number in scientific notation is therefore \(8 \times 10^{-7}\).
Key Concepts
Writing Numbers in Scientific NotationIdentifying Decimal PointUnderstanding Exponents
Writing Numbers in Scientific Notation
When we write numbers in scientific notation, we are using a system designed to express very large or very small numbers succinctly and with clear precision. This method is particularly useful in the sciences, where exact quantities are essential, but the numbers can get unwieldy.
The basic concept is to rewrite the number as the product of two factors - the first being a digit or a combination of a digit and decimal that's greater than or equal to 1 but less than 10, and the second being a power of 10. This neatly packages the number regardless of how many zeroes follow or precede it.
For instance, consider the number 0.0000008 from our exercise. To convert this to scientific notation, we need to make it into a number between 1 and 10 multiplied by 10 raised to an appropriate exponent. The precise steps might involve moving the decimal point and considering the direction of such movements relative to its exponent, as we'll delve in further sections.
The basic concept is to rewrite the number as the product of two factors - the first being a digit or a combination of a digit and decimal that's greater than or equal to 1 but less than 10, and the second being a power of 10. This neatly packages the number regardless of how many zeroes follow or precede it.
For instance, consider the number 0.0000008 from our exercise. To convert this to scientific notation, we need to make it into a number between 1 and 10 multiplied by 10 raised to an appropriate exponent. The precise steps might involve moving the decimal point and considering the direction of such movements relative to its exponent, as we'll delve in further sections.
Identifying Decimal Point
The decimal point plays a crucial role in understanding and manipulating numbers, especially when converting to scientific notation. It marks the boundary between whole numbers and fractional values in our base-10 system. When we're dealing with scientific notation, we focus on where the decimal point is and where it needs to go.
In our example, the number 0.0000008 has its decimal point at the very start, to the right of the initial zero. To convert it to scientific notation, we need to move this decimal point so that it comes after the first non-zero digit - which, in this case, is 8. Moving the decimal point to the right makes our number 8, now we just need the correct power of 10 to match the original value. This action of shifting the decimal point is directly linked to the exponent used in the scientific notation, a concept we explain next.
In our example, the number 0.0000008 has its decimal point at the very start, to the right of the initial zero. To convert it to scientific notation, we need to move this decimal point so that it comes after the first non-zero digit - which, in this case, is 8. Moving the decimal point to the right makes our number 8, now we just need the correct power of 10 to match the original value. This action of shifting the decimal point is directly linked to the exponent used in the scientific notation, a concept we explain next.
Understanding Exponents
Exponents are shorthand for repeated multiplication and form an integral part of scientific notation. In the context of scientific notation, the exponent on the number 10 tells us how many places the decimal point has been moved from its original position.
If the original number was less than one (like our 0.0000008 example), the exponent will be negative, indicating that the decimal point moved to the right. Conversely, if the number is greater than one, the exponent is positive, as the decimal point moves to the left.
To get from 0.0000008 to 8, you move the decimal 7 places to the right. Therefore, the exponent in the expression of scientific notation becomes -7 because it shows how much smaller the number needs to be to return to its original value. Thus, our final expression is written as \(8 \times 10^{-7}\), effectively summarizing the original number in a concise, scientific format.
If the original number was less than one (like our 0.0000008 example), the exponent will be negative, indicating that the decimal point moved to the right. Conversely, if the number is greater than one, the exponent is positive, as the decimal point moves to the left.
To get from 0.0000008 to 8, you move the decimal 7 places to the right. Therefore, the exponent in the expression of scientific notation becomes -7 because it shows how much smaller the number needs to be to return to its original value. Thus, our final expression is written as \(8 \times 10^{-7}\), effectively summarizing the original number in a concise, scientific format.
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