Problem 87 - Algebra and Trigonometry

Question

Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $$ \left(3 \times 10^{4}\right)\left(2.1 \times 10^{3}\right) $$

Step-by-Step Solution

Answer

The result of the multiplication is \(6.3 \times 10^{7}\).

Step 1: Multiply Decimal Components

First, multiply the decimal components of the two numbers. This means multiplying 3 and 2.1, which equals 6.3.

Step 2: Add the Powers of 10

Next, add the powers of 10. This will be the sum of 4 and 3, which gives 7.

Step 3: Write the Result in Scientific Notation

The result of the multiplication in scientific notation is achieved by placing the result of the decimal multiplication in front of \(x 10^{n}\), where \(n\) is the sum of the powers. Therefore, the final result is \(6.3 \times 10^{7}\).

Key Concepts

Multiplying Powers of 10

When dealing with scientific notation, multiplying numbers with powers of 10 involves simple arithmetic with the exponents. Understanding how to manage these powers makes the process efficient and straightforward. Scientific notation expresses numbers in the form of \( a \times 10^{n} \), where \( a \) is a decimal number, and \( 10^{n} \) represents its magnitude.When multiplying powers of 10:

• Keep the power base constant, which is 10.
• Add the exponents together.

This is because of the rule \( a^{m} \times a^{n} = a^{m+n} \).For example, multiplying \( 10^{4} \) by \( 10^{3} \) requires adding 4 and 3, resulting in \( 10^{7} \). Therefore, the new expression maintains the constant base and is simplified to a single power of 10.

Decimal Multiplication

Decimal multiplication refers to the process of multiplying numbers that are not whole. When you multiply large numbers expressed in scientific notation, start with their decimal components.For the example \( (3 \times 10^{4})(2.1 \times 10^{3}) \):

• Multiply the two decimal numbers: 3 and 2.1.
• This yields a result of 6.3.

To perform decimal multiplication efficiently,

• Position the numbers correctly aligning decimal points.
• Ignore the decimal points at first, multiply like whole numbers.
• Count decimal places of both numbers for final placement in the product.

Here, the result is 6.3, maintaining its determination for further combination with powers of 10.

Exponent Addition

Exponent addition is a calculation pivotal when dealing with numbers in scientific notation. Adding exponents involves straightforward arithmetic, crucial in combining powers of 10.In our example:

• Identify the exponents: 4 from \( 10^{4} \) and 3 from \( 10^{3} \).
• Simply add these exponents: 4 + 3.
• This produces a new exponent of 7.

These steps result from the properties of exponents which dictate that like bases being multiplied lead to their powers being summed. Hence, multiplying expressions like \( 10^{4} \) and \( 10^{3} \) results in \( 10^{7} \). Ultimately, adding exponents helps in consolidating multiple exponential expressions into one clear and simplified term.