Problem 173 - Introductory Chemistry Atoms First

Question

Write each number in scientific notation, taking all trailing zeros to be nonsignificant: (a) 502,000 (b) \(0.000038402\) (c) \(436,000,000\) (d) 8470 (e) \(0.00591\) (f) \(0.658\)

Step-by-Step Solution

Answer

The numbers in scientific notation are: (a) \(5.02 \times 10^5\) (b) \(3.8402 \times 10^{-5}\) (c) \(4.36 \times 10^8\) (d) \(8.47 \times 10^3\) (e) \(5.91 \times 10^{-3}\) (f) \(6.58 \times 10^{-1}\)

Step 1: (a) Convert 502,000 to scientific notation.

First, identify the number between 1 and 10 that we will multiply by a power of 10. In this case, it's 5.02. Next, determine the power of 10 by counting the number of decimal places we move to get from 502,000 to 5.02. Since we moved the decimal place 5 times to the left, our power of 10 would be 10^5. Therefore, the scientific notation for 502,000 is \(5.02 \times 10^5\).

Step 2: (b) Convert 0.000038402 to scientific notation.

The number between 1 and 10 that we will multiply by a power of 10 is 3.8402. To get from 0.000038402 to 3.8402, we need to move the decimal place 5 times to the right. This makes our power of 10 to be 10^-5. So, the scientific notation for 0.000038402 is \(3.8402 \times 10^{-5}\).

Step 3: (c) Convert 436,000,000 to scientific notation.

For this number, the number between 1 and 10 that we will multiply by a power of 10 is 4.36. We move the decimal place 8 times to the left to get from 436,000,000 to 4.36, making the power of 10 as 10^8. The scientific notation for 436,000,000 is \(4.36 \times 10^8\).

Step 4: (d) Convert 8470 to scientific notation.

The number between 1 and 10 is 8.47. To get from 8470 to 8.47, we need to move the decimal place 3 times to the left. This gives us a power of 10^3. The scientific notation for 8470 is \(8.47 \times 10^3\).

Step 5: (e) Convert 0.00591 to scientific notation.

For this number, the number between 1 and 10 that we will multiply by a power of 10 is 5.91. We move the decimal place 3 times to the right to get from 0.00591 to 5.91, making the power of 10 as 10^-3. The scientific notation for 0.00591 is \(5.91 \times 10^{-3}\).

Step 6: (f) Convert 0.658 to scientific notation.

In this case, the number between 1 and 10 is 6.58. We only need to move the decimal place 1 time to the right to get from 0.658 to 6.58, so our power of 10 is 10^-1. The scientific notation for 0.658 is \(6.58 \times 10^{-1}\).

Key Concepts

Trailing Zeros

In scientific notation, when working with numbers like 502,000 or 436,000,000, we often encounter trailing zeros. Trailing zeros are simply zeroes at the end of a number that do not add any value to its magnitude. These zeros are often not considered significant figures.
This means in situations where you are tasked with converting a number to scientific notation and have to consider trailing zeros to be nonsignificant, you can ignore them while determining the first part of the number, which should be between 1 and 10.

• For 502,000, ignoring the trailing zeros gives us 5.02 as our base number.
• For 436,000,000, similarly, we focus on 4.36.

Converting these into scientific notation simply involves pairing these base numbers with their respective powers of 10.

Decimal Places

Understanding decimal places is key when converting numbers into scientific notation. Decimal places tell us how many places a decimal has been moved from its original position to transform the number between 1 and 10.
Let's consider the example of 0.000038402, where we move the decimal five places to the right to transform it into 3.8402. This movement is represented as multiplying by a power of 10. The power of 10 becomes negative when moving the decimal to the right, indicating a smaller number. Thus, 0.000038402 becomes

• 3.8402 multiplied by \(10^{-5}\)

For 502,000 where we move the decimal five places to the left, the exponent is positive, leading us to

• 5.02 multiplied by \(10^{5}\)

Power of 10

The power of 10 in scientific notation indicates how many times you need to move the decimal place to convert the number into a more manageable form that is between 1 and 10. Whether the exponent is positive or negative depends on the direction you move the decimal.

• Positive powers of 10, such as \(10^3\) in 8470, mean the decimal is moved to the left to increase number size.
• Negative powers, like \(10^{-1}\) in 0.658, indicate moving the decimal to the right for smaller numbers.

Understanding the power of 10 helps simplify calculations and communicate large numbers efficiently.

Nonsignificant Figures

Nonsignificant figures are those that do not impact the numerical value's precision in the context of your calculations. In the exercise, nonsignificant figures often refer to trailing zeros, which were excluded from the significant digits in scientific notation. This means for numbers like 8470, only significant digits influence the base number. For example,

• 8.47 becomes the primary focus when determining its scientific notation.

Understanding which figures are nonsignificant helps you concentrate on what truly alters the computation and number representation. This allows for cleaner, more efficient calculation and representation in scientific notation.