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Chapter 5: Problem 63

Express each number in scientific notation. 32,000

Short Answer

Expert verified
The number 32000 can be expressed in scientific notation as \(3.2 \times 10^4\).

Step by step solution

01

Identify The Given Number

The given number here is 32000. Our objective is to express this in scientific notation.
02

Express In Scientific Notation

The number 32000 can be expressed as \(3.2 \times 10^4\) in scientific notation. This is because the decimal in the number 32000 is moved 4 places to the left to make the number 3.2, which is a number between 1 and 10. It is then multiplied with \(10^4\) (or 10000) to get back the original number, 32000.

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Key Concepts

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

Decimal Point Movement
In scientific notation, moving the decimal point is a crucial step. When converting a whole number like 32,000, start by identifying the location of the decimal point. In whole numbers, it is located after the last digit, even though it's not usually shown. Hence, for 32,000, it's right after the zero.

To express 32,000 in scientific notation, the goal is to shift the decimal point so that the number is between 1 and 10. For 32,000, this means you move the decimal point four places to the left, turning it into 3.2. This process effectively reduces the size of the number while keeping its value the same when paired with the correct power of ten.
Powers of Ten
Understanding powers of ten is essential when learning scientific notation. A power of ten is a way of expressing repeated multiplication of the number ten by itself. For example,
  • \(10^1 = 10\)
  • \(10^2 = 100\)
  • \(10^3 = 1,000\)
The exponent, or power of the ten, represents the number of times ten is multiplied by itself.

In scientific notation, moving the decimal point in the number creates a need to multiply by a corresponding power of ten. For 32,000 becoming 3.2, since the decimal point was moved four places to the left, the power of ten used is \(10^4\). This means multiplying 3.2 by 10,000 to return to the original number.
Standard Form
Scientific notation converts numbers into a standard form where they are easily readable and usable in calculations. The standard form is written as \(a \times 10^n\),where:
  • \(a\) is a digit between 1 and 10, which is achieved by moving the decimal point.
  • \(10^n\) is the power of ten that scales the number back to its original value.
In the case of 32,000, once you've shifted the decimal point, you use the power of ten to write the number in standard form: \(3.2 \times 10^4\). This format simplifies large numbers, making them easier to work with in scientific and general calculations.

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