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Chapter 1: Problem 72

List two advantages of using scientific notation. Answers may vary.

Short Answer

Expert verified
Scientific notation simplifies calculations and makes numbers easier to read and write.

Step by step solution

01

- Understanding Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form by using powers of 10. For example, the number 4,500 can be written as \(4.5 \times 10^3\) in scientific notation.
02

- Advantage 1: Simplifies Calculations

One advantage of using scientific notation is that it simplifies calculations. When dealing with large numbers like \(6.022 \times 10^{23}\) (Avogadro's number), scientific notation makes it easier to perform multiplication and division, as you can add or subtract the exponents.
03

- Advantage 2: Easier to Read and Write

Another advantage is that scientific notation makes very large or very small numbers easier to read and write. For instance, instead of writing 0.00000000123, you can write it as \(1.23 \times 10^{-9}\), which is more concise and reduces the likelihood of errors.

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Key Concepts

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

Simplifies Calculations
Scientific notation helps in making complex calculations more manageable. Imagine you have to multiply very large numbers, like the mass of an electron, which is about 9.11 脳 10^-31 kg, with Avogadro's number, 6.022 脳 10^23. Calculating this without scientific notation would be cumbersome due to numerous zeroes.
With scientific notation:
  • Multiply the significant figures (9.11 脳 6.022).
  • Add the exponents (-31 + 23).
Resulting in approximately 54.86 脳 10^-8, or more precisely, 5.486 脳 10^-7.
In scientific form, calculations involve simply managing exponents and significant figures, making large and small number operations easier and reducing errors. This method greatly benefits fields like physics and chemistry, where such numbers are common.
Easier to Read and Write
Scientific notation also makes numbers easier to read and write. Large numbers like the radius of the Earth (about 6,371,000 meters) and tiny numbers like the size of an atom (approximately 0.0000000001 meters) can be quite daunting.
Instead, using scientific notation, these become more concise:
  • The Earth's radius: 6.371 脳 10^6 meters.
  • The size of an atom: 1 脳 10^-10 meters.
This notation reduces cognitive load, making it quicker to interpret and less prone to mistakes. It's especially useful in data representation and scientific papers, where space and clarity are crucial.
Powers of 10
The backbone of scientific notation is the use of powers of 10. This method relies on breaking down numbers into a coefficient between 1 and 10, multiplied by 10 raised to an exponent.
For example:
  • 1,000,000 is 1 脳 10^6.
  • 0.0001 is 1 脳 10^-4.
Utilizing powers of 10 makes it easier to understand the scale of numbers. A positive exponent indicates a large number, whereas a negative exponent suggests a very small one.
Powers of 10 serve as a universal language in science and engineering, aiding in communication and comprehension. They help in representing far-ranging metrics from astronomical distances to microscopic measurements, consistently and clearly.

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Most popular questions from this chapter

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ 19-\left(4+2 \cdot 3^{2}\right) $$

Use set-builder notation to write each set. \(\\{24,26,28,30,32\\}\)

Translate to an algebraic expression. Three times the sum of the cubes of two numbers

Write an equivalent expression using a commutative law. Answers may vary. $$ (7 x) y $$

Write an equivalent expression using an associative law. $$ -7(a b) $$
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