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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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