Chapter 0: Problem 133
In Problems 129-136, write each number as a decimal. \(1.1 \times 10^{8}\)
Short Answer
Expert verified
110,000,000
Step by step solution
01
Understand Scientific Notation
Scientific notation is a way to express very large or very small numbers. It is written as the product of a number (between 1 and 10) and a power of 10.
02
Identify the Components
In the expression \(1.1 \times 10^{8}\), the number 1.1 is the coefficient, and \(10^{8}\) is the power of 10.
03
Interpret the Power of 10
The power of 10, \(10^{8}\), means that the decimal point in the coefficient (1.1) needs to be moved 8 places to the right.
04
Move the Decimal Point
Starting with 1.1, move the decimal point 8 places to the right. This will add 7 zeros after the 1 and shift the decimal to the end.
05
Write the Decimal Number
After moving the decimal point 8 places to the right, the number becomes 110,000,000 (one hundred ten million).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Decimal Conversion
Decimal conversion is the process of converting numbers, often from scientific notation, into standard decimal form. Scientific notation is handy for expressing very large or very small numbers in a more compact form.
For example, in the scientific notation expression of \(1.1 \times 10^8\), it's essential to understand how to convert it into a standard decimal form.
### Step-by-Step Conversion
1. Start with the coefficient, in this case, 1.1.
2. Determine the power of 10, which in this case is 8.
3. Move the decimal point in the coefficient 8 places to the right. This is done because the power of 10 is positive. If it were negative, you would move the decimal point to the left.
4. As you move the decimal, you will add zeros as necessary. For 1.1, moving the decimal 8 places to the right means you end up with 110,000,000.
So, during conversion, always follow these steps, ensuring the number you end up with accurately reflects the intended value in the standard decimal form.
For example, in the scientific notation expression of \(1.1 \times 10^8\), it's essential to understand how to convert it into a standard decimal form.
### Step-by-Step Conversion
1. Start with the coefficient, in this case, 1.1.
2. Determine the power of 10, which in this case is 8.
3. Move the decimal point in the coefficient 8 places to the right. This is done because the power of 10 is positive. If it were negative, you would move the decimal point to the left.
4. As you move the decimal, you will add zeros as necessary. For 1.1, moving the decimal 8 places to the right means you end up with 110,000,000.
So, during conversion, always follow these steps, ensuring the number you end up with accurately reflects the intended value in the standard decimal form.
Powers of 10
Understanding the powers of 10 is crucial for working with scientific notation. The power of 10 tells you how many places to move the decimal point.
### What is a Power of 10?
Powers of 10 are exponents of the number 10. For example, \(10^1\) is 10, \(10^2\) is 100, and so forth. If the exponent is positive, it indicates a multiplication: \(10^3 = 10 \times 10 \times 10 = 1000\). If the exponent is negative, it indicates a division: \(10^{-3} = \frac{1}{10 \times 10 \times 10} = 0.001\).
### Applying to Decimal Movement
In the scientific notation expression of \(1.1 \times 10^8\), the exponent 8 shows that you move the decimal point in 1.1 eight places to the right. This is because \(10^8\) equals 100,000,000, essentially making 1.1 much larger.
Conversely, if the expression was \(1.1 \times 10^{-8}\), you would move the decimal point 8 places to the left, making it a very small number. This would indicate that you are working with tiny values.
Knowing how to work with powers of 10 simplifies the process of dealing with both very large and very small numbers.
### What is a Power of 10?
Powers of 10 are exponents of the number 10. For example, \(10^1\) is 10, \(10^2\) is 100, and so forth. If the exponent is positive, it indicates a multiplication: \(10^3 = 10 \times 10 \times 10 = 1000\). If the exponent is negative, it indicates a division: \(10^{-3} = \frac{1}{10 \times 10 \times 10} = 0.001\).
### Applying to Decimal Movement
In the scientific notation expression of \(1.1 \times 10^8\), the exponent 8 shows that you move the decimal point in 1.1 eight places to the right. This is because \(10^8\) equals 100,000,000, essentially making 1.1 much larger.
Conversely, if the expression was \(1.1 \times 10^{-8}\), you would move the decimal point 8 places to the left, making it a very small number. This would indicate that you are working with tiny values.
Knowing how to work with powers of 10 simplifies the process of dealing with both very large and very small numbers.
Large Numbers
When dealing with very large numbers, scientific notation becomes extremely useful. With scientific notation, we can express these large values in a more manageable form.
### Examples of Large Numbers
1. The distance from the Earth to the Sun is about 93,000,000 miles. In scientific notation, it’s written as \(9.3 \times 10^7\).
2. The national debt of some countries can be in the trillions. For example, 20 trillion dollars can be written as \(2 \times 10^{13}\).
### Significance of Large Numbers
When solving problems or presenting data, working with large numbers in scientific notation helps reduce errors and makes the calculations easier to understand. By converting from scientific notation to decimal form, like turning \(1.1 \times 10^8 \) into 110,000,000, one can more readily grasp the concept of vast magnitudes, crucial in subjects such as astronomy, economics, and engineering.
In summary, mastering the conversion from scientific notation to normal decimal form and understanding the relevance of large numbers are pivotal for performing accurate and comprehensible calculations in both academic and real-world contexts.
### Examples of Large Numbers
1. The distance from the Earth to the Sun is about 93,000,000 miles. In scientific notation, it’s written as \(9.3 \times 10^7\).
2. The national debt of some countries can be in the trillions. For example, 20 trillion dollars can be written as \(2 \times 10^{13}\).
### Significance of Large Numbers
When solving problems or presenting data, working with large numbers in scientific notation helps reduce errors and makes the calculations easier to understand. By converting from scientific notation to decimal form, like turning \(1.1 \times 10^8 \) into 110,000,000, one can more readily grasp the concept of vast magnitudes, crucial in subjects such as astronomy, economics, and engineering.
In summary, mastering the conversion from scientific notation to normal decimal form and understanding the relevance of large numbers are pivotal for performing accurate and comprehensible calculations in both academic and real-world contexts.
