Problem 106 - Algebra A Combined Function

Question

A beam of light travels \(9.460 \times 10^{12}\) kilometers per year. How far does light travel in 10,000 years? Write the result in scientific notation.

Step-by-Step Solution

Answer

Light travels \(9.460 \times 10^{16}\) kilometers in 10,000 years.

Step 1: Understanding the Problem

The problem asks us to determine the distance that light travels over a period of 10,000 years. We are given that light travels \(9.460 \times 10^{12}\) kilometers in one year.

Step 2: Calculate the Distance

To find out how far light travels in 10,000 years, we need to multiply the distance that light travels in one year by 10,000. This can be represented by the equation: \[ \text{Distance in } 10,000 \text{ years} = 9.460 \times 10^{12} \times 10,000 \]

Step 3: Multiplying with Scientific Notation

Rewrite 10,000 in scientific notation as \(1 \times 10^4\). The expression becomes: \[ 9.460 \times 10^{12} \times 1 \times 10^4 = 9.460 \times 10^{12 + 4} \]

Step 4: Simplify the Expression

Combine the exponents to simplify the expression: \[ 9.460 \times 10^{16} \]

Step 5: Final Verification

Ensure that the result \(9.460 \times 10^{16}\) is properly formatted in scientific notation, which it is.

Key Concepts

Light Year Calculation

A light year is the distance that light travels in a vacuum in one year. Light years are used in astronomy to measure the vast distances between stars and galaxies. To calculate how far light travels over any number of years, you multiply the distance light travels in one year by the number of those years.
In our example, we know that light travels a distance of \(9.460 \times 10^{12}\) kilometers in one year. Thus, if we want to find out how far it travels in 10,000 years, we multiply the distance for one year by 10,000. This forms the mathematical basis for light year calculations.
Here's a simple breakdown:

• Identify the distance light travels in one year.
• Determine the number of years for which you need the distance calculated.
• Multiply these two values.

Multiplication in Scientific Notation

Scientific notation is a way of writing numbers that are too big or too small to be conveniently written in decimal form. It uses powers of ten, helping to represent large numbers more succinctly. In our exercise, both numbers in the multiplication are in scientific notation: \(9.460 \times 10^{12}\) kilometers per year, and we rewrite 10,000 as \(1 \times 10^4\).
Here's how you approach multiplication:

• Write each number in scientific notation if not already.
• Multiply the significant digits (those in front of the power of ten).
• Keep the base 10 the same, but add the exponents.

This allows you to keep calculations manageable while dealing with very large numbers.

Exponent Addition

When multiplying numbers in scientific notation, adding the exponents is a key step. Since our numbers are both forms of powers of 10, this process simplifies the overall multiplication. For example, in our problem:
We have \(10^{12} \) from the distance light travels in one year, and \(10^4\) from rewriting 10,000 in scientific notation. To multiply these, we add their exponents: \(12 + 4 = 16\).
Here’s a quick guide:

• Ensure each number is expressed as a digit times a power of ten.
• Keep the same base (which is 10).
• Add the exponents together.

Thus, the result in scientific notation is \(9.460 \times 10^{16}\). This step is crucial in ensuring our answer remains succinct and easy to interpret, maintaining its scientific notation integrity.