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Chapter 1: Problem 11

Convert the following values from scientific notation to standard notation: (a) \(4.885 \times 10^{4} \mathrm{mg}\) (b) \(8.3 \times 10^{-6} \mathrm{~m}\) (c) \(4.00 \times 10^{-2} \mathrm{~m}\)

Short Answer

Expert verified
(a) 48850 mg, (b) 0.0000083 m, (c) 0.04 m.

Step by step solution

01

Understand Scientific Notation

Scientific notation is a way to express very large or very small numbers conveniently. A number in scientific notation is generally written as a product of a coefficient and a power of ten, such as \(a \times 10^n\), where \(1 \leq a < 10\) and \(n\) is an integer.
02

Convert (a) from Scientific to Standard Notation

For \(4.885 \times 10^4\), the power of 10 indicates the number of decimal places to shift the coefficient. Since the power is 4, shift the decimal 4 places to the right: \(4.885\) becomes \(48850\, \mathrm{mg}\).
03

Convert (b) from Scientific to Standard Notation

For \(8.3 \times 10^{-6}\), the power of 10 is negative, so shift the decimal 6 places to the left. Add zeroes as placeholders: \(8.3\) becomes \(0.0000083\, \mathrm{m}\).
04

Convert (c) from Scientific to Standard Notation

For \(4.00 \times 10^{-2}\), shift the decimal 2 places to the left due to the negative power: \(4.00\) becomes \(0.04\, \mathrm{m}\).

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Key Concepts

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

Standard Notation Conversion
Standard notation is a way of writing numbers without the exponents that are part of scientific notation. In standard notation, numbers appear in their everyday form, meaning that all decimal places are either shifted to the right or left according to the value of an exponent.
Let's look at how conversion occurs:
  • When converting from scientific to standard notation, first identify the power of ten. For example, if you start with \(4.885 \times 10^{4}\), the exponent is \(4\).
  • The exponent tells you how many places to move the decimal point. A positive exponent, such as \(4\), means you shift the decimal point to the right. In this case, \(4.885\) becomes \(48850\).
  • Conversely, if the exponent is negative, as in \(8.3 \times 10^{-6}\), shift the decimal to the left. Here, it results in \(0.0000083\).
  • Always add zeros as placeholders while shifting the decimal to maintain the number's value.
Exponents in Scientific Notation
The exponent in scientific notation signifies how many times the base, which is typically 10, is multiplied by itself. This property makes scientific notation very useful for writing extremely large or small numbers in a more manageable form.
Here鈥檚 how exponents function:
  • An exponent of \(4\) in \(4.885 \times 10^{4}\) indicates that \(10\) should be raised to the power of \(4\), leading to a multiplication factor of \(10000\).
  • A negative exponent, such as \(-6\) in \(8.3 \times 10^{-6}\), shows that \(10\) should be divided into 1, yielding \(\frac{1}{1000000}\), which greatly reduces the number.
  • The exponent can dictate a considerable change in the magnitude of the number, making it particularly effective for scientific calculations where precision over a broad range of values is essential.
Decimal Place Shifting
Decimal place shifting is crucial when converting between scientific and standard notation.
Here鈥檚 how this mechanism works:
  • The number of places the decimal moves corresponds directly to the exponent in scientific notation.
  • For positive exponents, move the decimal to the right. For instance, in \(4.885 \times 10^{4}\), shift the decimal 4 places right, changing it into \(48850\).
  • For negative exponents, move the decimal to the left. So, \(8.3 \times 10^{-6}\) changes to \(0.0000083\) when the decimal shifts 6 places left.
  • It鈥檚 helpful to count each digit's shift and use zeros as necessary to ensure the number's magnitude remains accurate.

Understanding how to shift the decimal correctly is a key skill in working with scientific notation.

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Most popular questions from this chapter

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