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Chapter 9: Problem 24

Write each number in scientific notation. $$0.001276$$

Short Answer

Expert verified
0.001276 in scientific notation is \(1.276 \times 10^{-3}\).

Step by step solution

01

Identify the Significant Digits

The number 0.001276 has four significant digits after the zeros: 1, 2, 7, and 6.
02

Move the Decimal Point

Move the decimal point to the right of the first non-zero digit. In this case, moving it three places to the right gives 1.276.
03

Determine the Exponent

Since we moved the decimal point three places to the right, the exponent of 10 will be negative three: \(-3\).
04

Write in Scientific Notation

Combine the significant digits with the power of 10 to write the number in scientific notation: \(1.276 \times 10^{-3}\).

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Key Concepts

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

Understanding Significant Digits
Significant digits are crucial in scientific notation as they reflect the precision of a number. In our example, the number 0.001276 has four significant digits: 1, 2, 7, and 6. To determine which digits are significant, focus on numbers that are different from zero, unless zeros are between non-zero digits or follow significant digits.
  • Zeros before the digit 1 in 0.001276 are not significant. They only indicate the position of the decimal point.
  • Every digit after the zeros, in this case, 1, 2, 7, and 6, is significant because they contribute to the accuracy and value of the number.
Identifying the right number of significant digits is essential because it tells you how accurately the number is measured or calculated. The way significant digits are managed affects the scientific notation and consequently influences any calculations and understanding derived from these figures.
Moving the Decimal Point
The decimal point is fundamental to converting a number into scientific notation. In scientific notation, the decimal point is moved to ensure the number is expressed as a value between 1 and 10.
  • For 0.001276, identify the first non-zero digit, which is 1.
  • To convert this into scientific notation, you move the decimal point three places to the right. This transforms 0.001276 into 1.276.
This ensures your number is properly scaled between 1 and 10. Moving the decimal effectively by the right number of places assures the accurate representation of the number's magnitude. It also simplifies comparison and calculations involving large or small numbers.
Exponent of 10 and its Importance
In scientific notation, the exponent of 10 reveals how far the decimal point was moved and in which direction. This exponent is the key to understanding the scale of the number after adjusting the decimal point.
  • Since the decimal point in 0.001276 was moved three places to the right, the exponent is negative, specifically -3.
  • This is written as 10 to the power of -3, indicating the small size of the initial number.
The exponent of 10 shows the factor you would multiply 1.276 by to return to the original number in standard form. The negativity of the exponent indicates that this is a number less than one. Recognizing how to calculate and apply this exponent is vital for conveying very large or very small numbers compactly and accurately in scientific notation.

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Most popular questions from this chapter

Complete the following table. $$\begin{array}{|c|}\hline \text { Number } & \text { Square } \\\\\hline x & x^2 \\\\\hline 1 & \\\\\hline 2 &\\\\\hline3 & \\\\\hline 4 & \\\\\hline 5 & \\\\\hline 6 & \\\\\hline 7 & \\\\\hline\end{array}$$

Multiply the following expressions. \(\left(6 x^{2} y^{3}\right)\left(-5 x^{3} y^{5}\right)\)

Find the value of each of the following expressions when \(x=4\). $$3 x-5$$

Simplify each expression, and write all answers in scientific notation. $$\frac{\left(3 \times 10^{7}\right)\left(8 \times 10^{4}\right)}{6 \times 10^{5}}$$

Multiply the following expressions. $$(x+2)(x+3)$$
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