Problem 17 - Elementary Algebra

Question

Perform each multiplication. $$ \left(3 \times 10^{5}\right)\left(2 \times 10^{12}\right) $$

Step-by-Step Solution

Answer

Question: Multiply the given numbers in scientific notation: (3 x 10^5) and (2 x 10^12). Answer: (3 x 10^5)(2 x 10^12) = 6 x 10^17.

Step 1: Multiply the coefficients

First, we multiply the coefficients (3 and 2) together: $$ 3 \times 2 = 6 $$

Step 2: Add the exponents

Now, we add the exponents (5 and 12) together: $$ 5+12 = 17 $$

Step 3: Combine the results

Finally, we combine the results from steps 1 and 2 to get the final answer in scientific notation: $$ \left(3 \times 10^{5}\right)\left(2 \times 10^{12}\right) = 6 \times 10^{17} $$

Key Concepts

Multiplication of Exponents

Multiplying exponents is a key concept when dealing with scientific notation. This concept is quite straightforward once you grasp the basic rule: you must add the exponents. In scientific notation, numbers are often represented as a product of a coefficient and a power of 10. For example, \(3 \times 10^5\) indicates a scientific number format.
When you multiply two numbers in scientific notation, like \((3 \times 10^5)(2 \times 10^{12})\), focus on the exponents of ten.
Here's the rule: when you multiply powers of the same base, you keep the base and add the exponents. In this example:

• The base is 10, from \(10^5\) and \(10^{12}\).
• Add the exponents: \(5 + 12\).
• The result is \(10^{17}\).

A helpful tip to remember is that multiplying powers means adding their exponents. This keeps the multiplication process simple and effective.

Coefficients

In scientific notation, coefficients are regular numbers that are multiplied by powers of 10. The coefficients in any scientific notation formula must be treated separately from the exponents. Here’s how to handle coefficients in multiplication.
Consider the coefficients in the example \((3 \times 10^5)\) and \((2 \times 10^{12})\). The coefficients here are 3 and 2.

• Multiply these coefficients together: \(3 \times 2 = 6\).

It’s important to note that these are multiplied just as you would with any normal numbers, without any regard to the exponents. Once done, this result is combined with that from the exponents.
Understanding coefficients allows you to properly form results in scientific notation after manipulating the exponents. Keep in mind, coefficients are always handled separately before combining both solutions.

Addition of Exponents

The process of adding exponents is what authorizes the magic of scientific notation. It revolves around the principle that when multiplying numbers with the same base, the exponents must be added. This understanding allows for the simplification of complex multiplication into a succinct form.
In our example of multiplying \( (3 \times 10^5)(2 \times 10^{12}) \), exponents "5" and "12" are apart of powers of 10, the base remains 10:

• Add the exponents \(5 + 12\) to get 17.

This addition of exponents directly results from the distributive properties of multiplication over addition.
Once you master this simple add-on process, you’ll find it much easier to express extremely large or small numbers using scientific notation. It ensures numbers remain manageable and easy to work with in various scientific and mathematical calculations.
Adding exponents simplifies handling large numbers, making computations less daunting.