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Chapter 5: Problem 104

Write each number in scientific notation. $$0.006$$

Short Answer

Expert verified
The number 0.006 in scientific notation is \(6.0 \times 10^{-3}\).

Step by step solution

01

Determine Decimal Movement

Examine the number 0.006. To obtain a number between 1 and 10, we need to move the decimal point three places to the right. So, the number becomes 6.0.
02

Determine Power of Ten

The decimal point was moved to the right, which indicates a negative power of 10. The number of steps taken will be the exponent, so the result will be multiplied by \(10^{-3}\).
03

Write in Scientific Notation

Combine the results from step 1 and 2 to express 0.006 in scientific notation. The number is now 6.0 and the power of ten is -3, therefore the number in scientific notation is \(6.0 \times 10^{-3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decimal Movement
In the process of converting a number like 0.006 into scientific notation, understanding decimal movement is critical. The goal is to transform the number so that it's expressed as a coefficient between 1 and 10. Here, we start with 0.006 and want to obtain a number within this range.
To do this, we move the decimal point. For 0.006, we shift the decimal three places to the right. As a result, the number changes to 6.0, which lies in the desired range.
  • Identify the original position of the decimal point.
  • Count how many places you move the decimal to reach a number between 1 and 10.
  • For the number 0.006, moving the decimal three places gives us 6.0.
This adjustment is essential for expressing small numbers clearly in scientific notation.
Power of Ten
Once you've moved the decimal to achieve a number like 6.0, next comes determining the power of ten. Every time you move the decimal, it affects the exponent in scientific notation.
For our example (0.006 becoming 6.0), since the decimal was moved three places to the right, it results in a negative power of ten. Each space shifted to the right indicates a decrease in the order of magnitude.
  • Track the direction of movement: right signifies a negative adjustment.
  • Count the steps moved; these become the exponent's value in the power of ten.
  • Here, three steps translate to an exponent of -3, depicted as 10-3.
Understanding the power of ten is fundamental for accurately expressing numbers in scientific notation.
Negative Exponent
In scientific notation, a negative exponent is closely linked to small decimal numbers. The negative sign indicates how many times the decimal must be moved to the right to return to the original number. It decreases as the number grows smaller.
Using 0.006, we previously found a new position of 6.0 through the decimal movement. Because we shifted three steps to the right, the scientific notation uses a negative exponent, forming 10-3. This denotes that the number 6.0 must be divided by 10 three times to revert to the original form.
  • Negative exponents are crucial for very small numbers.
  • A change of place to the right equals a negative power alteration.
  • Our case, 0.006, becomes 6.0 脳 10-3 in scientific notation.
This negative adjustment reflects the true scale of the original number.

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