Problem 130 Solve. Write the answers using s... [FREE SOLUTION] (2024)

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Chapter 5: Problem 130

Solve. Write the answers using scientific notation.Explain why each of the following is not scientific notation: $12.6 \times10^{8} ; 4.8 \times 10^{1.7} ; 0.207 \times 10^{-5}$

Short Answer

Expert verified

12.6 is not between 1 and 10; 1.7 is not an integer; 0.207 is less than 1.

Step by step solution

Identify valid scientific notation

In scientific notation, a number is expressed as the product of two factors: a coefficient and 10 raised to a power. The coefficient must be a number greater than or equal to 1 and less than 10. The exponent must be an integer.

Analyze the first example

Consider the expression: $12.6 \times 10^{8}$. The coefficient here is 12.6, which is not between 1 and 10. Therefore, this is not proper scientific notation.

Analyze the second example

Consider the expression: $4.8 \times 10^{1.7}$. The exponent in scientific notation must be an integer. Since 1.7 is not an integer, this expression is not in scientific notation.

Analyze the third example

Consider the expression: $0.207 \times 10^{-5}$. The coefficient here is 0.207, which is less than 1. Therefore, this expression is not in scientific notation.

Fix the first example

To convert $12.6 \times 10^{8}$ to scientific notation, adjust the coefficient to be between 1 and 10. This can be done by converting it to $1.26 \times 10^{9}$.

Fix the second example

To convert $4.8 \times 10^{1.7}$ to scientific notation, change the exponent to the nearest integer and adjust the coefficient if necessary. In this case, conflict arises as scientific notation does not typically round exponents. Thus, correctly, it should be an integer value such as $4.8 \times 10^{2}$ for an approximate scale.

Fix the third example

To convert $0.207 \times 10^{-5}$ to scientific notation, adjust the coefficient to be between 1 and 10. This can be done by converting it to $2.07 \times 10^{-6}$.

Key Concepts

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

coefficients in scientific notation

Scientific notation is a way to express very large or very small numbers in a concise form. It involves a coefficient and an exponent. The coefficient in scientific notation must always be a number greater than or equal to 1 and less than 10.
For example, in the number 3.5 x 10^4, the coefficient is 3.5. Notice how it falls within the range of 1 to 10.
To better understand why coefficients must fall within this range, consider the number 12.6 x 10^8 from the exercise. Here, 12.6 is not within the required range. Instead, we convert it to 1.26 x 10^9. This adjustment ensures that the coefficient is valid while maintaining the value's scale.
By adhering to this rule, scientific notation remains standardized, making it easier for everyone to interpret values correctly.

integer exponents

In scientific notation, the exponent must be an integer. This means it must be a whole number that can be positive, negative, or zero.
Let's consider the example in the exercise where the number 4.8 x 10^1.7 is given. Here, the exponent is 1.7, which is not an integer. Scientific notation does not allow for decimal exponents, so this form is invalid.
When dealing with exponents in scientific notation, always make sure to use whole numbers. If you encounter a decimal exponent like 1.7, you should adjust it to the nearest whole number or find another way to represent the value based on context. For example, rounding 1.7 to 2 results in 4.8 x 10^2. However, note that this rounding approach is for illustrative purposes and proper practice involves retaining integer precision.
By using integer exponents, we preserve the simplicity and standardization of scientific notation. This makes it universally understood and easily manageable in mathematical computations.

adjusting scientific notation

Sometimes, we encounter values that need to be adjusted to fit into proper scientific notation. This typically involves changing the coefficient or the exponent.
Consider the example from the exercise: 0.207 x 10^-5. Here, the coefficient 0.207 is not within the range of 1 to 10. To adjust this to proper scientific notation, we modify the coefficient to 2.07, and the exponent to -6, giving us 2.07 x 10^-6.
These adjustments ensure that both the coefficient and exponent comply with scientific notation rules. Here's a simple process for adjustment:

Identify if the coefficient is not between 1 and 10.
Adjust the coefficient by moving the decimal point.
Compensate for this shift by altering the exponent accordingly. Moving the decimal to the left increases the exponent, while moving it to the right decreases the exponent.

By following these steps, you can easily convert any number into proper scientific notation. This keeps the expressions standardized and easy to interpret, aiding in the clarity and accuracy of our mathematical work.

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Most popular questions from this chapter

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