Problem 3 One astronomical unit is about \... [FREE SOLUTION]

Chapter 1: Problem 3

One astronomical unit is about \(1.5 \times 10^{8} \mathrm{km}\). Explain why this is the same as \(150 \times 10^{6} \mathrm{km}\)

Short Answer

Expert verified

By converting \(1.5 \times 10^8 \text{km}\) to \(150 \times 10^6 \text{km}\), we see they are equivalent.

Step by step solution

Understanding Scientific Notation

Scientific notation is a way to express very large or very small numbers conveniently. It is written as the product of a coefficient and a power of 10. For example, in the number \(1.5 \times 10^{8}\), 1.5 is the coefficient and \(10^{8}\) is the power of 10.

Converting Notation

To show that \(1.5 \times 10^{8} \text{km}\) is the same as \(150 \times 10^{6} \text{km}\), we need to convert one form into the other. Notice that \(10^8 = 10^2 \times 10^6\). Therefore, \(1.5 \times 10^8\) can be rewritten as \(1.5 \times 10^2 \times 10^6\).

Simplifying the Expression

Now, calculate \(1.5 \times 10^2\): \(1.5 \times 10^2 = 1.5 \times 100 = 150\). Therefore, \(1.5 \times 10^8\) is equivalent to \(150 \times 10^6\).

Conclusion

Now we have shown that \(1.5 \times 10^8 \text{km}\) is indeed the same as \(150 \times 10^6 \text{km}\) by converting and simplifying the notation.

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Key Concepts

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

Astronomical Unit

An astronomical unit (AU) is a standard unit of measurement used in astronomy to describe distances in space. It is defined as the average distance between the Earth and the Sun.

One astronomical unit is approximately equal to 149.6 million kilometers (149,600,000 km). For simplicity, this is often rounded to about 150 million kilometers or, in scientific notation, 1.5 脳 10^8 km.

Using the astronomical unit helps astronomers and scientists communicate vast distances more easily instead of using very large numbers. For instance, instead of saying a planet is 448,000,000 km from the sun, one could say it is about 3 AU away.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number (the base) to the power of an exponent. For example, in the expression 10^8, 10 is the base, and 8 is the exponent, indicating that 10 is multiplied by itself 8 times.

In scientific notation, exponentiation is used to express very large or very small numbers. This helps in making calculations more manageable and communication more clear.

For instance:

10^3 = 10 脳 10 脳 10 = 1,000
10^-3 = 1/10 脳 1/10 脳 1/10 = 0.001
1.5 脳 10^8 means 1.5 is multiplied by 10 seven more times

Exponentiation is fundamental in scientific notation as it allows us to represent numbers concisely.

Conversion of Notation

Conversion of notation is the process of changing a number from one form of scientific notation to another.

Let's take the example provided:
1.5 脳 10^8 km and 150 脳 10^6 km.
To show these are equivalent, we start by converting 10^8.
Using the fact that 10^8 = 10^2 脳 10^6, we can rewrite 1.5 脳 10^8 as: 1.5 脳 10^2 脳 10^6.

Next, compute 1.5 脳 10^2:

1.5 脳 10^2 = 1.5 脳 100 = 150

Now we have 150 脳 10^6.

Therefore, we have successfully shown that 1.5 脳 10^8 km is equivalent to 150 脳 10^6 km by simple conversion and calculation. Such conversions are often necessary in scientific work to facilitate comparison and understanding of large-scale measurements.

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91影视

One astronomical unit is about \(1.5 \times 10^{8} \mathrm{km}\). Explain why this is the same as \(150 \times 10^{6} \mathrm{km}\)

Short Answer

Step by step solution

Understanding Scientific Notation

Converting Notation

Simplifying the Expression

Conclusion

Key Concepts

Astronomical Unit

Exponentiation

Conversion of Notation

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