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Chapter 7: Problem 25

Convert each number into scientific notation. $$47.5$$

Short Answer

Expert verified
4.75 \( \times 10^1 \).

Step by step solution

01

- Identify the coefficient

The coefficient is a number between 1 and 10 obtained by moving the decimal point in the original number. For 47.5, move the decimal point one place to the left to get 4.75.
02

- Determine the exponent

The exponent is determined by the number of places you moved the decimal point. For 47.5, the decimal point moved one place to the left, so the exponent is 1.
03

- Combine to form scientific notation

Combine the coefficient and the exponent to get the number in scientific notation. Therefore, 47.5 in scientific notation is \[4.75 \times 10^1\].

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Key Concepts

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

Coefficient
In scientific notation, the coefficient is a crucial part. It's a number between 1 and 10. For example, to convert 47.5 into scientific notation, we first need to find the coefficient.

To do this, we move the decimal point in the original number so that only one non-zero digit is to the left of the decimal point. In our case, moving the decimal point one place to the left transforms 47.5 into 4.75.

This new number, 4.75, is our coefficient. The process of determining the coefficient helps us manage large or small numbers more conveniently in scientific notation.
Exponent
Once we have identified the coefficient, the next step is to determine the exponent.

The exponent tells us how many places we moved the decimal point. It also indicates the power of 10 that we need to multiply the coefficient by to get back to the original number.

In our example, we moved the decimal point one place to the left to turn 47.5 into 4.75. Therefore, the exponent is 1. We write this exponent as \(10^1\).

By combining the coefficient and the exponent, we can represent the original number in a concise and standardized way.
Decimal Point
The position of the decimal point determines both the coefficient and the exponent in scientific notation.

Initially, the decimal point in our number 47.5 is between the 7 and the 5. To convert it into scientific notation, we move the decimal point one place to the left, making the number 4.75.

This movement is important because it dictates the value of the exponent. If we move the decimal to the right instead, we would adjust the exponent accordingly.

The key is to ensure the coefficient remains a number between 1 and 10. Keeping track of the decimal point's original and new positions helps us correctly convert any number into scientific notation.

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Most popular questions from this chapter

In this section we defined \(a^{0}=1 .\) It is important to realize that a definition is neither right nor wrong-it just is. The proper question to ask about a definition is "Is it useful?" What happens if someone decides on an "alternative definition" such as \(a^{0}=7\) because 7 happens to be his or her favorite number? What happens when we consider the expression \(a^{0} \cdot a^{4} ?\) If you use exponent rule 1 you get \(a^{0} \cdot a^{4}=a^{0+4}=a^{4} .\) However, if you use this alternative definition, you get \(a^{0} \cdot a^{4} \stackrel{\underline{2}}{=} 7 a^{4} .\) The answer we get from the alternative definition is not consistent with the answer we get from the exponent rule. The exponent rules and this alternative definition cannot coexist. Since we do not want to throw away all the exponent rules, we must modify the alternative definition so that it is consistent with all the exponent rules. Verify that our definition of \(a^{0}=1\) is consistent with all five exponent rules.

A car rental agency charges \(\$ 15\) per day plus \(\$ 0.20\) per mile. If a 3 -day rental cost \(\$ 72,\) how many miles were driven?

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(2 a+5 y)^{2}$$

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(2 y+5)(y-3)$$

Solve each of the following problems algebraically. Be sure to label what the variable represents. A large, full tank contains several liquids. If one fifth of the tank is water, 5 gallons is orange juice, and three quarters of the tank is wine, how many gallons of wine are in the tank?
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