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Chapter 4: Problem 21

Write each number in standard notation. $$ 2.54 \times 10^{-3} $$

Short Answer

Expert verified
0.00254

Step by step solution

01

Identify the components of the scientific notation

The given number is in scientific notation: a = 2.54 b = -3
02

Understand the meaning of the exponent

The exponent \(-3\) indicates how many places to move the decimal point to the left. A negative exponent means the number is less than 1.
03

Move the decimal point

Starting from 2.54, move the decimal point 3 places to the left: 2.54 --> 0.254 --> 0.0254 --> 0.00254

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Key Concepts

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

Standard Notation
Scientific notation is a way to express very large or very small numbers. It simplifies these numbers using powers of 10. To convert a number from scientific notation to standard notation, you need to move the decimal point.
For example, the scientific notation: \(2.54 \times 10^{-3} \) means we have the number 2.54, and the exponent \(-3\) tells us to move the decimal point 3 places to the left.
This process involves:
  • Identify the number in scientific notation (2.54 in this case).
  • Note the exponent (\(-3\)), which shows the direction and number of places to move the decimal point.
  • Move the decimal point accordingly. Here, moving 3 places to the left results in 0.00254.
Decimal Points
Decimal points are crucial for representing non-whole numbers. They indicate fractions of a whole number.
In our example, 2.54 has a decimal point between the 2 and the 5, representing 2 whole units and 54 hundredths.
To convert from scientific to standard notation, the decimal point determines how the number changes:
  • Positive exponents move the decimal point to the right, making the number larger.
  • Negative exponents move the decimal point to the left, making the number smaller.
For \(2.54 \times 10^{-3}\), the exponent of \(-3\) means moving the decimal point 3 places left of 2.54, yielding 0.00254.
Remembering how decimal points function helps accurately translate between scientific and standard notation.
Negative Exponent
Exponents indicate how many times a number is multiplied by itself. A negative exponent specifically involves dividing by that same number.
With scientific notation, a negative exponent tells us to divide by a power of 10.
For example: \(10^{-3} = \frac{1}{10^3}\). This equals \frac{1}{1000}\ or 0.001.
In scientific notation, \2.54 \times 10^{-3}\):
  • 2.54 is our base number.
  • Negative exponent \(-3) informs us to move the decimal point 3 places to the left.
  • Converting to standard notation gives us 0.00254.
Mastering negative exponents is essential for interpreting and converting scientific notation accurately.
It makes complex calculations more manageable.

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Most popular questions from this chapter

Solve each problem. In 2017 , the population of Costa Rica was approximately \(4.93 \times 10^{6} .\) The population density was 96.6 people per square kilometer. (Data from The World Almanac and Book of Facts.) (a) Write the population density in scientific notation. (b) To the nearest square kilometer, what is the area of Costa Rica?

Use the FOIL method to find each product. $$ (z-w)(3 z+4 w) $$

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The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ (0.3 x-1.6 y)^{2} $$

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