Problem 85 - Elementary Algebra

Question

Find the product for the following problems. Write the result in scientific notation. $$ \left(8 \times 10^{-3}\right)\left(3 \times 10^{-6}\right) $$

Step-by-Step Solution

Answer

Question: Multiply (8 x 10^(-3)) and (3 x 10^(-6)) and present the result in scientific notation. Answer: 24 x 10^(-9)

Step 1: Multiply the Mantissas

Multiply the mantissas (the numbers before the exponent terms) of the given numbers: $$ 8 \times 3 = 24 $$

Step 2: Add the Exponents

Add the exponents of the given numbers: $$ (-3) + (-6) = -9 $$

Step 3: Combine the Mantissa and Exponent

Combine the result from Step 1 and Step 2 to present the result in scientific notation: $$ 24 \times 10^{-9} $$ The product of the two given numbers in scientific notation is: $$ \left(8 \times 10^{-3}\right)\left(3 \times 10^{-6}\right) = 24 \times 10^{-9} $$

Key Concepts

Multiply Mantissas

When dealing with scientific notation, the numbers you typically encounter have a form that involves a base number, called the mantissa, and an exponent associated with a power of ten. In simpler terms, these numbers look something like this: \( a \times 10^n \). Here, \( a \) is the mantissa and \( n \) is the exponent. To multiply numbers expressed in scientific notation, you start by multiplying the mantissas.

Let's take an example from the exercise: \( 8 \times 10^{-3} \) and \( 3 \times 10^{-6} \). The mantissas here are 8 and 3. Start by multiplying these mantissas:

• \( 8 \times 3 = 24 \)

When you multiply the mantissas, you're dealing with the whole numbers or decimal numbers before the power of ten. It's very straightforward—just like a regular multiplication you would do with any two integers or decimals.

Add Exponents

Once you have multiplied the mantissas, the next step is to deal with the exponents. Exponents in scientific notation indicate how many times the base number (10 in this case) is used in multiplication. When multiplying numbers in scientific notation, you don't multiply the exponents—instead, you add them together. This is because the base of the exponents is the same (10), and in mathematical terms, \( 10^a \times 10^b = 10^{a+b} \).

For our example, the numbers \( 8 \times 10^{-3} \) and \( 3 \times 10^{-6} \) have exponents \(-3\) and \(-6\), respectively. You add these exponents:

• \(-3 + -6 = -9\)

Remember, when adding negative exponents, add them algebraically as you would with any signed numbers. This allows your product to stay within the realm of scientific notation.

Product in Scientific Notation

The final step in finding the product in scientific notation is to combine the results from the previous steps. You've already multiplied the mantissas to obtain 24 and added the exponents to get -9. The next thing is to express these results as a single number in scientific notation.

To do this, you combine the mantissa and the new exponent:

• \( 24 \times 10^{-9} \)

This notation shows the product of the original numbers. You have the mantissa, 24, which is a straightforward whole number, and you pair it with the exponent, \(-9\), which denotes the power of ten by which 24 is multiplied. This means 24 is scaled down by nine decimal places due to the negative exponent.

Keep in mind, if your final mantissa is not between 1 and 10, you might have to adjust it to fit the standard scientific notation form. In this case, no adjustment is needed as 24 fits perfectly within acceptable parameters for many applications.