Question

Write the number in decimal notation.Number of Air Sacs in the Lungs: \(3.5 \times 10^{8}\)

Short answer

The decimal notation of the number \(3.5 \times 10^{8}\) is \(350,000,000\).

Step-by-step solution

Identify the Base Number and Power

The number given is in scientific notation, where \(3.5\) is the base number and \(8\) is the exponent to base \(10\).

Convert to Decimal Notation

To convert to decimal notation, we move the decimal point to the right, 8 places, filling the necessary places with zeros. This is because the exponent of \(10\) is \(8\) and a positive exponent means we move the decimal point to the right.

Write the Final Answer

After moving the decimal 8 places to the right, the number now reads as \(350,000,000\)

Key concepts

Decimal Notation

Decimal notation is a way of expressing numbers that uses a decimal point to separate the whole part from the fractional part of a number. This system, also known as base-10 notation, is the most commonly used system of numbers and is familiar in everyday mathematics. In the example given, converting from scientific to decimal notation involves expanding the number to show its full value without exponents.

When converting a number in scientific notation like the number of air sacs in the lungs, which is expressed as \(3.5 \times 10^8\), we need to understand that the exponent indicates how many places we move the decimal point. Since the exponent in this case is positive, we move the decimal point to the right. For every number place we move over, we insert a zero if necessary, until we have shifted eight places, resulting in the expansive form of \(350,000,000\). Decimal notation makes it easier to visualize and compare numbers without the use of exponents.

Exponents

Exponents, also referred to as powers, are a shorthand way to denote repeated multiplication of a number by itself. The exponent you see in a number tells you how many times to multiply the base number by itself. In our exercise, the exponent is \(8\), which means the base number, \(3.5\), must be multiplied by 10 a total of 8 times.

An exponent is typically written as a superscript to the right of the base number. When an exponent is positive, as in \(10^8\), we understand that the number is large and the decimal point is moved to the right when converting to decimal notation. Conversely, a negative exponent would mean the opposite: the number is small, and the decimal point would be moved to the left. Knowing how to work with exponents is fundamental to understanding scientific notation and performing operations on numbers in exponential form.

Base Number

The concept of a 'base number' in mathematics refers to the number that is used as the starting point for exponential multiplication. In scientific notation, the base number often follows a decimal point and is typically a number between 1 and 10, including decimals. In the given exercise, the base number is \(3.5\). This base number is multiplied by the power of 10 raised to the specified exponent.

Understanding the base number is essential because it represents the significant figures of the number in question. These are the numbers that convey the precision of the measurement related to the number of air sacs in the lungs. The base number, along with the exponent, gives us a precise and compact way to represent very large or very small numbers, which is much more practical and less error-prone than writing out a number with many zeros.