Problem 89 - Elementary Algebra

Question

Find the product for the following problems. Write the result in scientific notation. $$ \left(1.06 \times 10^{-16}\right)\left(2.815 \times 10^{-12}\right) $$

Step-by-Step Solution

Answer

Question: Find the product of the given numbers in scientific notation: $(1.06 \times 10^{-16})(2.815 \times 10^{-12})$. Answer: The product of the given numbers in scientific notation is $2.9859 \times 10^{-28}$.

Step 1: Multiply the coefficients

First, we'll multiply the coefficients 1.06 and 2.815. This gives us: $$ (1.06)(2.815) = 2.9859 $$

Step 2: Add the exponents

Now, we'll add the exponents: -16 and -12. This gives us the following: $$ (-16) + (-12) = -28 $$

Step 3: Write the result in scientific notation

Finally, we'll combine the product of the coefficients and the sum of the exponents to write the result in scientific notation: $$ 2.9859 \times 10^{-28} $$ So, the product of the given numbers in scientific notation is: $$ \left(1.06 \times 10^{-16}\right)\left(2.815 \times 10^{-12}\right) = 2.9859 \times 10^{-28} $$

Key Concepts

Multiplying Exponents

Exponents allow us to express numbers in a compressed form, making calculations more manageable. When multiplying numbers in scientific notation, the rule for exponents is super helpful. The basic rule is to add the exponents together. For instance, if you have terms like \(10^{-16}\) and \(10^{-12}\), the exponents are added as follows:

• \((-16) + (-12) = -28\)

This rule works because the base of the exponents, which is 10, remains the same. You just need to focus on adding the powers (or exponents) together. Keep this rule in mind whenever you're working with scientific notation.

Coefficients

Coefficients are the numbers in front of the exponents when you're dealing with scientific notation. They work like regular numbers and are multiplied in the usual way. In our problem, the coefficients are 1.06 and 2.815. Here's what you do:

• Multiply 1.06 by 2.815 to get 2.9859

Coefficients are crucial because they determine the number part of the scientific notation. Make sure you multiply them correctly just as you would in basic multiplication, and then carry their product forward when writing your final answer in scientific notation.

Basic Algebra

Basic algebra involves operations such as addition, subtraction, multiplication, and division. When dealing with scientific notation, basic algebra helps us manage the coefficients and the exponents separately. Breaking the problem into smaller steps, like multiplying the coefficients and handling the exponents as a separate step, simplifies the process.

• First, handle the coefficients: 1.06 × 2.815
• Next, manage the exponents: \((-16) + (-12)\)

With these steps, you can solve the problem methodically. By applying basic algebra, you can tackle each part of the scientific notation without feeling overwhelmed.

Product

The product is the result of multiplying two numbers together. In the context of scientific notation, it includes both the multiplication of coefficients and the effective combining of exponents. By finding the product of our example problem:

• Multiply the coefficients: 2.9859
• Combine the exponents: \(10^{-28}\)

So, the final product expressed in scientific notation is \(2.9859 \times 10^{-28}\). Remember, the product is always the combination of both components in scientific notation - the coefficient multiplied and the exponents added.