Problem 81 - Elementary Algebra

Question

Find the product for the following problems. Write the result in scientific notation. $$ \left(3 \times 10^{6}\right)\left(7 \times 10^{7}\right) $$

Step-by-Step Solution

Answer

Answer: The product of the given numbers in scientific notation is \(21 \times 10^{13}\).

Step 1: Multiply the coefficients

Multiply the coefficients, which are the numbers before the power of 10. In this case, it's 3 and 7: $$ 3 \times 7 = 21 $$

Step 2: Add the exponents of the base 10

Add the exponents of the base 10 from both numbers: $$ (10^{6}) \times (10^{7}) = 10^{6+7} = 10^{13} $$

Step 3: Combine the results

We combine the results obtained in the previous two steps to get the final answer in scientific notation: $$ (21) (10^{13}) $$ So, the product of the given numbers in scientific notation is: $$ 21 \times 10^{13} $$

Key Concepts

Multiplying Coefficients

When you encounter problems involving scientific notation, the first step is often to multiply the coefficients. Coefficients are simply the numbers that appear before the powers of 10 in scientific notation. To multiply coefficients, you use regular arithmetic multiplication. For example, if you have \(3 \times 10^6\) and \(7 \times 10^7\), you would focus on the 3 and the 7.
\[3 \times 7 = 21\]
This multiplication is straightforward and doesn't involve any special rules beyond those you would use in normal arithmetic.

Adding Exponents

Once you have multiplied the coefficients, the next step in scientific notation multiplication is to manage the exponents. When multiplying numbers in scientific notation, if the base is the same (in most cases, the base is 10), you simply add the exponents together. This step is a basic algebraic operation known as the 'Product of Powers Property'.
\[(10^6) \times (10^7) = 10^{6+7} = 10^{13}\]
Keep in mind that this rule only applies to multiplication and to exponents with the same base. It's a common mistake to try to apply this rule to addition or subtraction of numbers in scientific notation, which is incorrect.

Algebraic Operations

Scientific notation involves several algebraic operations, with multiplication and addition being the most common. Understanding how to manipulate exponents is key to performing these operations correctly. After multiplying the coefficients and adding the exponents, you combine these results to express the final answer in scientific notation. If the multiplication of coefficients results in a number not between 1 and 10, you would further adjust the result to maintain proper scientific notation format by moving the decimal point and adjusting the exponent accordingly.
Remember, scientific notation is used to simplify calculations and expressions, especially with very large or very small numbers, by keeping numbers concise and manageable.

Scientific Notation Format

The format of scientific notation is to express numbers as a product of two factors: a coefficient that is at least 1 but less than 10, and a power of 10. The exercise provided required you to multiply two numbers already in scientific notation, but it's essential to recall that the final answer must also be in the correct format.
\[21 \times 10^{13}\]
In this case, the coefficient is 21, which is not between 1 and 10. The correct scientific notation would be to divide the coefficient by 10 and increase the exponent by 1, resulting in:
\[2.1 \times 10^{14}\]
By ensuring the final product is written in proper scientific notation format, you demonstrate a clear understanding of the significance and purpose of this system of notation.