Question

It is roughly \(93,000,000\) miles from Earth to the sun. A scientist would express this number as A. \(93,000,000\) miles. B. \(93 \times 10^6\) miles. C. \(9.3 \times 10^7\) miles. D. \(0.0093 \times 10^9\) miles.

Short answer

The answer is C. 9.3 × 10^7 miles.

Step-by-step solution

Understand the Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is written as the product of a number (between 1 and 10) and a power of 10.

Convert to Scientific Notation Form

Express the given distance, 93,000,000 miles, in scientific notation form. Move the decimal point to the left until only one non-zero digit remains to the left of the decimal. 93,000,000 becomes 9.3 when the decimal is moved 7 places to the left.

Count the Number of Decimal Places Moved

Since the decimal point was moved 7 places to the left, the exponent will be +7. Thus, 93,000,000 miles is expressed as 9.3 × 10^7.

Identify the Correct Option

Compare the scientific notation obtained with the given options:A. 93,000,000 milesB. 93 × 10^6 milesC. 9.3 × 10^7 milesD. 0.0093 × 10^9 miles. The correct option is C.

Key concepts

decimal form

Decimal form is a way of writing numbers that uses a decimal point to separate the whole number part from the fractional part. For example, the number 93,000,000 in decimal form is simply 93,000,000. It's straightforward but can get cumbersome when dealing with very large or very small numbers.

To make large numbers easier to handle, we use scientific notation. This allows us to express numbers succinctly while retaining their full value.

In the given exercise, we start with 93,000,000 in its decimal form. As we convert this number to scientific notation, we simplify it, making it easier to understand and work with.

power of 10

Powers of 10 are a fundamental part of scientific notation. They help simplify the process of writing very large or small numbers.

When you multiply a number by a power of 10, you shift the decimal point to the right for positive exponents or to the left for negative exponents. For instance, moving the decimal to the left by 7 places for the number 93,000,000 gives us 9.3. Hence, 93,000,000 written in scientific notation is 9.3 multiplied by 10 raised to the power of 7, which is expressed as 9.3 × 10^7.

Specifically, for our exercise, converting 93,000,000 into scientific notation involves recognizing that we've moved the decimal point 7 positions to the left. This exponent, +7, indicates how many times 10 must be multiplied by itself to return to the original number.

expressing large numbers

Expressing large numbers using scientific notation is crucial when dealing with astronomy, physics, and other sciences. It allows scientists to write very large or very small numbers in a compact and manageable form.

For example, the distance from Earth to the sun, 93,000,000 miles, can be expressed as 9.3 × 10^7 in scientific notation. This notation tells us that 9.3 should be multiplied by 10 seven times, which gives the original distance.

Using scientific notation, we avoid writing long strings of zeros, and it becomes easier to perform mathematical operations like multiplication and division. It's a standard method in science and engineering because it simplifies data representation and calculation.