Problem 82 - Elementary Algebra

Question

Find the product for the following problems. Write the result in scientific notation. $$ \left(2 \times 10^{14}\right)\left(8 \times 10^{19}\right) $$

Step-by-Step Solution

Answer

Answer: The product of the two numbers is 16 × 10^33.

Step 1: Multiply the numbers

First, let's multiply the numbers (the coefficients) in the parentheses: $$(2 \times 8)$$

Step 2: Multiply the powers of ten

Next, we'll multiply the powers of ten. When multiplying powers of ten with the same base, we add the exponents: $$10^{14} \times 10^{19} = 10^{14 + 19}$$

Step 3: Combine the results

Now, we combine the results from Step 1 and Step 2: $$(2 \times 8)(10^{14 + 19})$$

Step 4: Calculate the product

Let's do the calculations: $$(2 \times 8)(10^{14 + 19}) = (16)(10^{33})$$

Step 5: Write the final result in scientific notation

Since the result \(16(10^{33})\) is already in scientific notation, the exercise is complete. So the final result is: $$ \left(2 \times 10^{14}\right)\left(8 \times 10^{19}\right) = 16 \times 10^{33} $$

Key Concepts

Multiplication of Powers

In mathematics, it’s crucial to understand how to handle powers, especially when they're involved in multiplication. When you multiply powers with the identical base, the rule is straightforward: add the exponents together.
This fundamental property simplifies complex expressions and calculations.
Here's how it works:

• Given two powers, say, \(a^m\) and \(a^n\), the product is \(a^{m+n}\).
• This rule specifically applies to powers with the same base, like powers of ten.

To visualize, let's take the example \(10^{14} \times 10^{19}\). Here, both numbers have a base of 10, so we simply add the exponents: \(10^{14 + 19} = 10^{33}\).
This addition translates a seemingly complex expression into something simpler, aligning perfectly for further computation or scientific notation.

Exponents Addition

Exponents provide a way to express repeated multiplication of a number by itself. Adding exponents is a key operation when multiplying like bases.
It’s essential to follow specific guidelines when handling them.
Let's delve deeper into this:

• With numbers sharing the same base, as in \(x^a \times x^b\), we add the exponents: \(x^{a+b}\).
• Background mathematics ensures this rule retains integrity across all number sets.
• This method condenses larger expressions into manageable calculations.

The step of adding exponents simplifies usage in equations, especially in scientific notation cases where vast numbers often encounter manipulation. This simplicity helps in maintaining accuracy and reducing errors in numerical computations, a mainstay of mathematical education.

Mathematics Education

Mathematics as a discipline holds principles rooted in logic and precision, viable tools in education methodologies. One of its essential aspects is forming a foundational understanding of concepts like scientific notation and powers.
In educational contexts, these topics bridge ordinary arithmetic with advanced fields like algebra and calculus.
Focusing on scientific notation:

• It represents numbers in a compact form \(a \times 10^n\), ideal for very large or small values.
• This notation eases complex calculations, frequently utilizing multiplication and exponents interactions.

Through mathematics education, students are equipped to move from basic multiplication to exploring the vast array of scientific concepts linked by these principles.
Ultimately, these skills are applicable in real-world problems, fostering critical thinking and problem-solving capabilities essential across disciplines.