Question

Write the number in scientific notation.Land Area of Earth: \(57,300,000\) square miles

Short answer

The land area of Earth in scientific notation is \(5.73 \times 10^7\) square miles.

Step-by-step solution

Identification

Identify the number we are given, which in this case is \(57,300,000\).

Conversion to scientific notation

To convert the given number into scientific notation, write it as a number between 1 and 10 multiplied by a power of 10. Here, the first step is to place the decimal point after the first digit, which makes the number \(5.73\). Then, count how many places the decimal point was moved. Here, it was moved 7 places to the left. Therefore, the exponent of 10 would be 7.

Final scientific notation

Now the given number can be written in scientific notation as \(5.73 \times 10^7\).

Key concepts

Converting to Scientific Notation

When working with very large or very small numbers, scientific notation is a convenient way to express them. It simplifies calculations and provides a standardized format for scientists and mathematicians.

To convert a number into scientific notation, you'll first need to rewrite the number so that it falls between 1 and 10. This is achieved by moving the decimal point to the right place. For example, with the land area of Earth at 57,300,000 square miles, you would rewrite this number as 5.73. The next step is critical: count how many places you've moved the decimal point. For the Earth's land area, the decimal point moves 7 places to the left, which means you'll use 10 raised to the power of 7, or \(10^7\), to compensate for this shift. The number then becomes \(5.73 \times 10^7\) in scientific notation.

In a nutshell, converting to scientific notation involves two key steps: adjusting the number to be between 1 and 10, and factoring the shift of the decimal place into the exponent of 10.

Properties of Exponents

Understanding the properties of exponents is critical when working with scientific notation. An exponent indicates the number of times a base is multiplied by itself. For instance, the expression \(10^7\) means that 10 is multiplied by itself 7 times.

The properties of exponents play a role when performing operations like multiplication and division. One key property is that to multiply powers with the same base, you add the exponents (e.g., \(10^2 \times 10^3 = 10^{2+3} = 10^5\)). Conversely, when dividing powers with the same base, you subtract the exponents (e.g., \(10^5 \/ 10^2 = 10^{5-2} = 10^3\)).

Another crucial property is the power of a power rule, which states that to raise a power to another power, you multiply the exponents (e.g., \((10^2)^3 = 10^{2 \times 3} = 10^6\)). These rules are fundamental when simplifying expressions in scientific notation and help maintain uniformity and accuracy in calculations.

Decimal Place Value

The concept of decimal place value is at the heart of the decimal numbering system, and consequently, it is equally important when converting numbers to scientific notation. Each position to the left or right of the decimal point represents a power of 10.

For example, in the number 57,300,000, the 5 is in the ten millions place, meaning it represents \(5 \times 10^7\). As you move to the right, each place value is 10 times smaller. So the next digit, 7, is in the millions place and represents \(7 \times 10^6\) and so on down to the ones place.

When converting to scientific notation, you're essentially finding the most significant digit's place value and representing the remaining digits as decimal fractions of that value. Therefore, understanding decimal place value is fundamental to accurately finding the correct exponent for the power of 10 when converting to and from scientific notation.