Problem 61 - Elementary and Intermediate Algebra

Question

Use scientific notation to perform the calculations. Give all answers in scientific notation and standard notation. \(\left(3.4 \times 10^{2}\right)\left(2.1 \times 10^{3}\right)\)

Step-by-Step Solution

Answer

In scientific notation: \(7.14 \times 10^5\). In standard notation: 714,000.

Step 1: Rearrange the Expression

First, we will rearrange the expression to separate the numbers and the powers of 10. The expression is \((3.4 \times 10^2)(2.1 \times 10^3)\). This can be rearranged as \((3.4 \times 2.1) \times (10^2 \times 10^3)\).

Step 2: Multiply the Numbers

Next, multiply the non-exponential parts of the expression: \(3.4 \times 2.1 = 7.14\).

Step 3: Apply the Laws of Exponents

Now, multiply the powers of ten. According to the laws of exponents, \(10^2 \times 10^3 = 10^{2+3} = 10^5\).

Step 4: Combine the Results

Combine the results from Steps 2 and 3: \(7.14 \times 10^5\). This is the result in scientific notation.

Step 5: Write the Standard Notation

To express the answer in standard notation, move the decimal point 5 places to the right: \(714,000\).

Key Concepts

Understanding the Laws of Exponents

The laws of exponents are fundamental rules that help simplify expressions involving powers. These rules make calculations faster and simpler, especially when dealing with large numbers. Here, we specifically used the law that states: when multiplying expressions with the same base, you add their exponents.
For example, when you multiply two exponential terms like \(10^2\) and \(10^3\), you keep the base (which is 10 in this case) and add the exponents together. You get:

• \(10^2 \times 10^3 = 10^{2+3} = 10^5\)

This rule helps in reducing the complexity and makes it easy to manage calculations involving large powers. Laws of exponents are not only applicable to the base 10, but they can also be applied to any other numerical base.
Remembering these simple rules can save time and ensure accuracy in your calculations.

Simplifying Multiplication of Exponents

When working with exponents, multiplication often crops up. Understanding how to handle it is essential. In our original exercise, after rearranging the terms, we dealt with two main parts: the non-exponential numbers and the powers of ten. To get a neat answer, you need to multiply these parts separately.

• First, multiply the standard numbers: \(3.4\) and \(2.1\), which gives us \(7.14\).
• Then, tackle the exponents by applying the laws of exponents as outlined in the previous section.

By processing each part individually, you simplify computations without making them cumbersome. This approach is particularly useful when dealing with numbers expressed in scientific notation. Mastering this skill ensures you can swiftly handle real-world problems involving large quantities or measurements efficiently.

From Scientific to Standard Notation

Translating numbers from scientific notation to standard notation is a handy skill. It makes it easier to understand and compare large numbers visually. Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10. In our example, the result \(7.14 \times 10^5\) is an elegant way to handle what could be a large number.

• To convert \(7.14 \times 10^5\) into standard notation, move the decimal point five places to the right. This shift translates our number to \(714,000\).

The number of spaces you move the decimal point corresponds to the power of ten. If the exponent is negative, the decimal point moves to the left instead, allowing for easy handling of both very large or very small numbers.
This method maintains precision and is a critical step in ensuring figures are correctly interpreted in both academic and practical situations.