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Chapter 5: Problem 83

Perform the indicated operation and express each answer in decimal notation. \(\left(4.1 \times 10^{2}\right)\left(3 \times 10^{-4}\right)\)

Short Answer

Expert verified
The answer is \(12.3 \times 10^{-2}\). In decimal notation, this is equivalent to 0.123.

Step by step solution

01

Multiply the Decimal Numbers

First, multiply the decimal numbers together. In this case, we multiply \(4.1\) and \(3\), which gives us \(12.3\).
02

Add the Exponents

Next, add the exponents together. In this case, we add \(2\) and \(-4\), which gives us \(-2\). So the 10's exponent after multiplication will be \(-2\).
03

Combine the Result

Finally, we combine the result from steps 1 and 2 to get the final answer. So, the answer is \(12.3\) times \(10^{-2}\).

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Key Concepts

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

Decimal Notation
Decimal notation is a way of representing numbers that uses a decimal point to separate the whole number from the fractional part. It is the standard form for writing numbers, particularly those that are not whole numbers. For example, numbers like 4.1, 3.5, and 12.3 are in decimal notation. This type of notation makes it easy to understand the size and value of a number, especially when dealing with non-integers.
  • The decimal point helps to differentiate between the integer part and the fractional part.
  • In the decimal number system, digits range from 0 to 9.
  • It's widely used in calculations because it's straightforward and familiar to most people.

In scientific calculations, we often encounter decimal numbers because they allow for high precision, which is essential in accurate measurements and calculations.
Exponent Rules
Exponent rules are guidelines that help us simplify operations involving powers of ten or any other base. Understanding these rules is crucial when working with scientific calculations, as they allow us to manipulate large or small numbers efficiently. The two basic rules are:
  • Multiplication Rule: When you multiply two powers of the same base, you add the exponents. For example, when multiplying \(10^2\) by \(10^{-4}\), you add the exponents: \(2 + (-4) = -2\).
  • Division Rule: When you divide two powers of the same base, you subtract the exponents.

These rules make calculations involving exponents much simpler by allowing us to express the result as another power of ten. This is particularly helpful in scientific notation, which we use to represent very large or very small numbers briefly and precisely.
Multiplying Decimals
Multiplying decimals may seem tricky at first, but it's simple once you understand the process. When multiplying decimal numbers, the steps are similar to multiplying whole numbers, with a few additional considerations for decimal places. Here鈥檚 how:
  • First, ignore the decimals and multiply the numbers as if they were whole numbers.
  • In the case of the given exercise, multiply 4.1 by 3 to get 12.3.
  • Count the total number of decimal places in both numbers you are multiplying. Here, 4.1 has one decimal place, and 3 has none, making it a total of one.
  • Place the decimal point in the product so that the number of decimal places is equal to the total you counted.

After multiplying decimals, always make sure to place the decimal correctly. This ensures that the product reflects the correct value, as seen in the exercise where 12.3 is the result of multiplying decimals.
Scientific Calculation
Scientific calculation often involves numbers that are either extremely large or incredibly small. Using scientific notation simplifies these calculations by representing numbers as a product of a number (between 1 and 10) and a power of ten. This method is efficient for expressing such data in a compact form.
  • Scientific Notation: It is typically written as \(a \times 10^n\), where a is a number greater or equal to 1 but less than 10, and n is an integer, representing the power of ten.
  • After performing calculations in scientific notation, we often convert the results back to decimal notation for readability.
  • In the given exercise, the result \(12.3 imes 10^{-2}\) needs to be expressed in decimal notation, which is 0.123.

Scientific calculation thus bridges the gap between complex mathematical operations and easily understandable results, facilitating both accurate computations and clear presentations.

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Most popular questions from this chapter

The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to \(2015 .\) Exercises 125-126 involve developing arithmetic sequences that model the data. In \(1990,18.4 \%\) of American women ages 25 and older had graduated from college. On average, this percentage has increased by approximately \(0.6\) each year. a. Write a formula for the \(n\)th term of the arithmetic sequence that models the percentage of American women ages 25 and older who had graduated from college \(n\) years after \(1989 .\) b. Use the model from part (a) to project the percentage of American women ages 25 and older who will be college graduates by 2029 .

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\sqrt{3}, 3,3 \sqrt{3}, 9, \ldots\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{30}\), when \(a_{1}=8000, r=-\frac{1}{2}\).

The sum, \(S_{n}\), of the first \(n\) terms of an arithmetic sequence is given by$$S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right),$$in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by$$S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r},$$in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(7,19,31,43, \ldots\)

Company A pays \(\$ 44,000\) yearly with raises of \(\$ 1600\) per year. Company B pays \(\$ 48,000\) yearly with raises of \(\$ 1000\) per year. Which company will pay more in year 10 ? How much more?
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