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Chapter 5: Problem 143

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

Short Answer

Expert verified
0.00254

Step by step solution

01

Understand Scientific Notation

In scientific notation, a number is written as the product of a decimal number and a power of 10. For example, in the expression \(2.54 \times 10^{-3}\), 2.54 is the decimal number and \(10^{-3}\) represents the power of 10.
02

Identify the Power of 10

In the given expression, the power of 10 is \(10^{-3}\). A negative exponent means that the decimal point moves to the left.
03

Shift the Decimal Point

Move the decimal point in the number 2.54 three places to the left because the exponent is -3. This means: 2.54 -> 0.254 -> 0.0254 -> 0.00254
04

Write the Number in Standard Notation

After moving the decimal point three places to the left, write the number in standard notation. It becomes 0.00254.

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Key Concepts

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

Standard Notation
Standard notation is the typical way of writing numbers without exponents. It's how you usually see numbers in everyday life. For example, instead of writing a number in scientific notation like \(2.54 \times 10^{-3}\), you would write it out as 0.00254. This is called converting from scientific notation to standard notation.
To convert a number from scientific notation to standard notation, you follow these steps:
- Identify the exponent associated with 10.
- If the exponent is negative, move the decimal point to the left. If positive, move it to the right.
Negative Exponents
In scientific notation, negative exponents tell us how many places to move the decimal point to the left. For instance, \( 10^{-3} \) means we move the decimal three places to the left.
Let's break down the example given in the exercise:
- The expression is \(2.54 \times 10^{-3}\).
- The negative exponent -3 means the decimal moves three places left. So, 2.54 becomes 0.00254.
Negative exponents can make large numbers look very small and vice versa. They are especially useful for dealing with very small values.
Decimal Point
The decimal point is crucial in both standard and scientific notation. It's the dot that separates the integer part of a number from its fractional part.
Moving the decimal point is the main action in converting scientific notation to standard notation.
- If a number has a negative exponent, move the decimal point to the left.
- If the exponent were positive, you would move it to the right.
Training yourself to quickly move the decimal point the correct number of places will make working with these forms of notation much easier. In our example, moving the decimal three places to the left transformed 2.54 into 0.00254.

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