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Chapter 2: Problem 1

Determine if each number is in scientific notation. \(7.23 \times 10^{5}\)

Short Answer

Expert verified
Yes, the number \(7.23 \times 10^{5}\) is in scientific notation, as it meets all three criteria: it has a non-zero digit before the decimal point, exactly one non-zero digit to the left of the decimal point, and is multiplied by a power of 10.

Step by step solution

01

Check the number before the decimal point

The number before the decimal point is 7, which is non-zero. So far, this number meets the criterion.
02

Check the number of non-zero digits to the left of the decimal point

There is exactly one non-zero digit to the left of the decimal point (7). The number meets this criterion as well.
03

Check if the number is multiplied by a power of 10

The given number is multiplied by \(10^{5}\), which is a power of 10. This criterion is also met.
04

Conclusion

Since the number \(7.23 \times 10^{5}\) meets all the criteria for scientific notation, we can conclude that it is indeed in scientific notation.

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Key Concepts

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

Power of 10
The concept of a "power of 10" is fundamental in scientific notation. It works with exponential expressions where 10 is raised to a certain power. For example, in the number \(7.23 \times 10^{5}\), "\(10^{5}\)" is the power of 10.
  • The exponent (5 in this case) tells us how many times to multiply 10 by itself.
  • A positive exponent, like 5, indicates multiplication, making the number larger.
  • A negative exponent would mean division, which would make the number smaller.
Understanding powers of 10 allows you to grasp large or small numbers easily in a compact form. This is why scientific notation is a preferred way to write very large numbers, like the distance to the sun, or very small numbers, such as the size of an atom. In our example, the power of 10, \(10^{5}\), implies that 7.23 is multiplied by 100,000.
Decimal Point
The decimal point is a key player when it comes to scientific notation. It separates the whole number from the fractional part. In the number \(7.23 \times 10^{5}\), the decimal point sits between the 7 and 2.
  • Correct placement is crucial; it affects the size of the number.
  • In scientific notation, there should always be one non-zero digit to the left of the decimal point.
  • The rest of the significant figures come after it.
Moving the decimal point influences the exponent in the power of 10. Each shift left increases the exponent by 1, while each shift to the right decreases it by 1. Placing the decimal correctly ensures the number is in the proper form, making it easy to work with, especially in mathematics and science fields.
Non-Zero Digit
In scientific notation, the significance of the non-zero digit can't be overstated. It is crucial for defining the standard form.
  • Only one non-zero digit should precede the decimal point.
  • This rule makes scientific notation compact and efficient.
  • The rest of the non-zero digits follow the decimal.
The non-zero digits help in maintaining the precision of the number. In the example \(7.23 \times 10^{5}\), the first non-zero digit is 7. Without it, the number wouldn't accurately express the same value. When dealing with real-world applications like measurements or statistical data, having precise and accessible values is essential.
Number Criterion
Scientific notation has specific criteria that must be met for a number to be considered correctly formatted:
  • The coefficient (number before the multiplication sign) must have exactly one non-zero digit before the decimal point.
  • The base must always be 10.
  • The exponent signifies how many times the base is used as a factor.
For a number like \(7.23 \times 10^{5}\) to be in scientific notation, it must fulfill all these conditions. Meeting each criterion ensures that the number stays manageable and standardized, making it easy to read and comprehend. This is particularly beneficial in fields that work with extreme values, ranging from astronomy to quantum physics. These criteria help in maintaining consistency across diverse applications.

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Most popular questions from this chapter

Simplify the expression using one of the power rules. $$\left(-5^{3}\right)^{2}$$

Simplify using the quotient rule. $$\frac{15 w^{2}}{w^{10}}$$

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Simplify using the quotient rule. $$\frac{6 v^{1} w}{54 v^{2} w^{-5}}$$

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