Problem 11 - Intermediate Algebra

Question

For Problems \(1-18\), write each of the following in scientific notation. \(0.347\)

Step-by-Step Solution

Answer

0.347 in scientific notation is \(3.47 \times 10^{-2}\).

Step 1: Identify the Significant Figures

The number 0.347 has three significant figures: 3, 4, and 7. In scientific notation, the aim is to express the number such that it has this same set of significant figures, starting with a digit from 1 to 9.

Step 2: Move the Decimal Point

To express 0.347 in scientific notation, move the decimal point to create a new number between 1 and 10. Move the decimal point right two places, changing it to 3.47.

Step 3: Determine the Power of Ten

Since we moved the decimal point two places to the right, the exponent will be negative. Therefore, the power of ten will be \(-2\).

Step 4: Write in Scientific Notation

Combine the number from Step 2, 3.47, with the power of ten from Step 3. The number 0.347 in scientific notation is written as \(3.47 \times 10^{-2}\).

Key Concepts

Significant Figures

Significant figures are all about precision in measurement or calculation. When you look at the number 0.347, the significant figures are the digits that carry meaning contributing to its measurement: 3, 4, and 7. In scientific notation, we preserve these significant figures to represent the number accurately. This is crucial because significant figures indicate how precise a measurement is. Here's what you should know:

• These figures include all non-zero digits, any zeros between them, and trailing zeros in the decimal part.
• When converting to scientific notation, significant figures ensure we maintain the number's true value and precision.

So, when 0.347 is expressed in scientific notation, its significant figures go into forming 3.47, which still holds those three important digits.

Decimal Point

The decimal point plays a crucial role in understanding where a number lies in relation to whole numbers and fractions. In 0.347, it separates the whole number from fractional parts:

• When converting a number to scientific notation, we shift the decimal to form a new number between 1 and 10.
• This results in the significant figures being positioned correctly for readability and calculation ease.

For instance, moving the decimal in 0.347 two places to the right, we get 3.47. This shift doesn't change the value inherently but adapts it for easier expression using exponents.

Exponents

Exponents in scientific notation are what express the magnitude increase or decrease from our original number to the new form. For the number 0.347, once we shift the decimal point:

• We determine the exponent based on how many places we moved the decimal.
• Moving the point two places to the right indicates a negative exponent of \(-2\).

So, in scientific notation, 0.347 becomes \(3.47 \times 10^{-2}\). The exponent of \(-2\) tells us that 3.47 is much smaller than the base unit, as reflected by the original decimal number.