Question

The distance from Earth to the Moon is \(2.39 \times 10^{5}\) miles, whereas the distance from the Moon to Mars is \(4.84 \times 10^{7}\) miles. What is the total distance from Earth to the Moon to Mars?

Short answer

The total distance is \(2.39 \times 10^5 + 4.84 \times 10^7\) miles.

Step-by-step solution

Understanding Scientific Notation

The scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. Here, the distances are given in scientific notation. For example, the distance from Earth to the Moon is given as \(2.39 \times 10^5\), meaning 2.39 times 10 raised to the power of 5.

Identifying the Given Values

We are given two distances: the distance from Earth to the Moon is \(2.39 \times 10^5\) miles, and the distance from the Moon to Mars is \(4.84 \times 10^7\) miles.

Adding Distances in Scientific Notation

To find the total distance from Earth to the Moon and then to Mars, we need to add the two distances. The total distance is: \(2.39 \times 10^5 + 4.84 \times 10^7\).

Converting to Same Power of 10 (Optional)

For addition, we could convert both terms to the same power of 10, but here we will handle them directly. We will align the smaller term by scaling it with the power of the larger one, if needed, but for this solution, we will not further simplify since they have different powers, and direct calculator input can be used.

Calculating the Total

Calculate the total distance using a calculator or by leaving the terms in symbolic form since they can't be directly simplified without additional information: \((2.39 \times 10^5) + (4.84 \times 10^7)\).

Conclusion

The total distance from Earth to the Moon to Mars is the sum in scientific notation. Since multiplication of powers is straightforward, convert the smaller term to a compatible form if required for simplification, otherwise address the summation using a calculator.

Key concepts

Addition of Distances

When it comes to finding the total distance between multiple points, such as Earth, the Moon, and Mars, we need to add the distances together. Distance addition is a fundamental concept in mathematics and in real-life applications where travel is involved.

In this exercise, you start with the distance from Earth to the Moon, which is given as \(2.39 \times 10^5\) miles. Then, you have the distance from the Moon to Mars, which is \(4.84 \times 10^7\) miles. These two distances must be added to find the total distance between Earth and Mars, via the Moon.

To add these distances, you may add them directly or choose to align them under the same power of 10, making them more manageable for calculation. However, due to the large difference in magnitude of their exponents, simply using a calculator might be more practical. Understanding these principles aids in executing accurate distance calculations, whether in planning a literal trip through space or solving math problems!

Distance Calculation

Calculating distances accurately is an essential mathematical skill, especially when dealing with observations of space or geography here on Earth. In this problem, we're tasked with calculating a journey's total distance from Earth, to the Moon, and onward to Mars.

To simplify, you break down the journey into segments. By identifying each individual distance, such as \(2.39 \times 10^5\) miles to the Moon, and a continued \(4.84 \times 10^7\) miles to Mars, you can address each portion individually before summing them up.

• Segment 1: Earth to Moon \( = 2.39 \times 10^5\)
• Segment 2: Moon to Mars \( = 4.84 \times 10^7\)

Adding these segments together will result in the full, collective distance for your trip. This process of summing segments ensures clarity and precision in distance calculations, particularly when using scientific notation to handle vast magnitudes.

Scientific Notation in Mathematics

Scientific notation is a shorthand method used in mathematics and science to write very large or very small numbers more conveniently. It's expressed as the product of two factors: a coefficient and 10 raised to an exponent.

For example, the distance from Earth to the Moon is written as \(2.39 \times 10^5\). Here, 2.39 is the coefficient, representing the significant digits, and \(10^5\) shows the scale of the number. This method allows for simplicity and precision when working with numbers that have many zeroes.

**Why Use Scientific Notation?**

• It makes reading and writing large numbers more efficient.
• It facilitates easy arithmetic operations, especially multiplication and division.
• It helps manage ranges of values quickly in scientific and engineering calculations.

In our universe, where distances can be astronomically high, such as to celestial bodies, scientific notation ensures we can understand and manipulate these figures without unnecessary complexity.