Problem 80 - Elementary Algebra

Question

Find the product for the following problems. Write the result in scientific notation. $$ \left(1 \times 10^{5}\right)\left(2 \times 10^{3}\right) $$

Step-by-Step Solution

Answer

Question: Multiply the two given numbers in scientific notation: \((1 \times 10^5)\) and \((2 \times 10^3)\). Answer: The product of the two numbers in scientific notation is \(\left(2 \times 10^{8}\right)\).

Step 1: Multiply the coefficients

We begin by multiplying the coefficients 1 and 2, which gives us: $$ 1 \times 2 = 2 $$

Step 2: Add the exponents

Next, we add the exponents together: $$ 5 + 3 = 8 $$

Step 3: Write the result in scientific notation

Now, we write the result in scientific notation, combining the values from steps 1 and 2: $$ \left(2 \times 10^{8}\right) $$ Therefore, the product of the two numbers given in scientific notation is \(\left(2 \times 10^{8}\right)\).

Key Concepts

Exponents Addition

When multiplying two numbers expressed in scientific notation, a crucial step is to add the exponents of the base 10 components together. This is a fundamental rule in exponentiation, which states that when multiplying powers with the same base, you keep the base and add the exponents. For example, when we have \(10^5\) and \(10^3\), the base here is 10, and the exponents are 5 and 3 respectively. According to the rule, you simply add the exponents: \[ 5 + 3 = 8 \].

The resulting exponent for the base 10 would then be 8, indicating that the multiplication has effectively 'shifted' the decimal point 8 places to the right when compared to a base unit. This process is reliably predictable, regardless of the size of the numbers involved, which makes scientific notation so valuable for handling extremely large or small numbers. Understanding this rule is pivotal as it empowers students to multiply numbers in scientific notation correctly and efficiently.

Multiplying Coefficients

As part of the multiplication process in scientific notation, multiplying the coefficients together is the first step. In our problem, the coefficients are 1 and 2. Multiplying these gives us \(1 \times 2 = 2\).

It's important to remember that the coefficients must be numbers between 1 and 10; if the multiplication of coefficients results in a number outside this range, the answer will need to be adjusted into proper scientific notation. However, with 1 and 2, the result is already within the acceptable range, so no further adjustments are required. Keeping the coefficients within this range ensures that the number remains correctly normalized, allowing for a consistent method of comparison between different scientific notations. Pay careful attention to the multiplication and normalization of coefficients, as this is key to maintaining the integrity of the scientific notation format.

Writing in Scientific Notation

Once you've multiplied the coefficients and added the exponents, you'll need to write the result in scientific notation. This involves placing the new coefficient in front of the multiplication sign and appending the base 10 raised to the newly calculated exponent. The final answer to our original problem is thus expressed as \(2 \times 10^8\).

Scientific notation follows a specific format where the coefficient is a number greater than or equal to 1 and less than 10, and the base is always 10. The power of 10 indicates the number of decimal places the decimal point must be moved to find the true value of the number. Writing in scientific notation is not just about achieving the correct answer; it's also important for ensuring clarity and precision, especially when dealing with very large or very small quantities in fields such as physics, chemistry, and engineering. Remember, proper scientific notation allows for easy comparison and manipulation of numerical values, as well as providing a standardized form that is universally understood in the scientific community.