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Chapter 7: Problem 32

Convert each number into scientific notation. $$234.8 \times 10^{-3}$$

Short Answer

Expert verified
2.348 \times 10^{-1}

Step by step solution

01

- Identify the Coefficient and Exponent

First, identify the given number and exponent in the expression. Here we have 234.8 as the number and \({10^{-3}}\) as the exponent.
02

- Convert Number to Scientific Form

Move the decimal point in 234.8 to make it a number between 1 and 10. This results in 2.348, and you moved the decimal 2 places to the left.
03

- Adjust the Exponent

Since the decimal was moved 2 places to the left, add 2 to the exponent to accommodate the shift. The new exponent becomes \(-3 + 2 = -1\).
04

- Write in Scientific Notation

Combine the coefficient 2.348 with the adjusted exponent. The final scientific notation is \(2.348 \times 10^{-1} \).

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Key Concepts

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

Coefficient
A coefficient is the number that is placed in front of a variable or another mathematical entity in an equation or an expression. In scientific notation, the coefficient is the number between 1 and 10 that we use to represent the significant figures of a quantity. For example, if we take the number 234.8, we move the decimal point to get 2.348, making 2.348 the coefficient. This step ensures we are simplifying the number while maintaining its value, which is crucial in scientific notation. Remember, the coefficient helps us convey large or small numbers more compactly.
Exponent
An exponent in scientific notation indicates how many times the base of 10 must be multiplied or divided by itself. Consider our number 234.8 with the exponent \(10^{-3}\). After moving the decimal point of 234.8 to get the coefficient 2.348, we need to adjust the exponent. Since we shifted the decimal point 2 places to the left, we increase the exponent by 2. Therefore, the exponent changes from -3 to \( -1 \). This result gives us \(2.348 \times 10^{-1}\). Exponents are crucial because they help us convert very large or very small numbers efficiently without changing their values.
Decimal Point
The decimal point is a dot used to separate the whole number part from the fractional part of a number. In scientific notation, the position of the decimal point is essential as it helps determine the coefficient. When converting a number like 234.8 into scientific notation, we must adjust the decimal point to ensure the coefficient is between 1 and 10. By moving the decimal point 2 places to the left, we change 234.8 to 2.348. This shift is a critical step because it allows us to rewrite the number as \(2.348 \times 10^{-1}\). Proper manipulation of the decimal point is needed to correctly apply the principles of scientific notation.

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