Question

Write each number in scientific notation. 32.14

Short answer

3.214 × 10^1

Step-by-step solution

Identify the Significant Figures

Determine the significant figures in the number. For 32.14, the significant figures are 3, 2, 1, and 4.

Place the Decimal Point

Rewrite the number so that there is only one non-zero digit to the left of the decimal point. Move the decimal point to between the 3 and 2, making it 3.214.

Determine the Exponent

Count how many places the decimal point was moved. The decimal was moved 1 place to the left.

Write in Scientific Notation

Combine the significant figures with the power of ten. The number 32.14 becomes 3.214 × 10^1.

Key concepts

significant figures

When working with numbers, significant figures (or 'sig figs') are crucial. They represent the digits in a number that carry meaningful information about its precision. For our example number, 32.14, we identify the significant figures: 3, 2, 1, and 4. These figures tell us how accurately the value is known.
Significant figures help in expressing numbers without overstating their precision. For instance, if the value were 32.140, the last zero would be significant only if it indicated an additional level of precision.
Knowing significant figures is vital in scientific notation since it ensures that you maintain the integrity and precision of the original number.

decimal point

The decimal point separates the whole number part from the fractional part of a number. It's crucial in scientific notation because you need to position it correctly to express numbers in a simplified form.
For the number 32.14, find a way to write it with only one non-zero digit to the left of the decimal point. We move the decimal point one place to the left so it sits between the 3 and 2, turning 32.14 into 3.214. This step ensures the number is in the proper format for scientific notation.

exponent

An exponent is a number that shows how many times a base number is multiplied by itself. In scientific notation, the exponent tells us how many places the decimal point was moved.
For 32.14, we moved the decimal point one place to the left. So, the exponent here is 1. This means that 3.214 in scientific notation is written as 3.214 × 10^1. The 10^1 indicates that 3.214 needs to be multiplied by 10 once to get back to the original number, 32.14.
Using exponents makes it easier to handle very large or very small numbers, simplifying calculations and making them more manageable.