Problem 23 - Elementary Algebra

Question

Find the value of \(\left(3 \times 10^{5}\right)\left(2 \times 10^{-2}\right)\).

Step-by-Step Solution

Answer

Answer: The product is 6 × 10^3.

Step 1: Write the numbers in scientific notation

Given: \((3 \times 10^{5})(2 \times 10^{-2})\)

Step 2: Multiply the coefficients

Multiply the coefficients (the numbers without the exponents): \(3 \times 2 = 6\)

Step 3: Use the exponent properties to multiply the powers of 10

When multiplying, we add the exponents: \(10^{5} \times 10^{-2} = 10^{5 + (-2)} = 10^3\)

Step 4: Combine the results

Combine the results from Step 2 and Step 3: \(6 \times 10^3\)

Step 5: Write the final answer

The value of \((3 \times 10^{5})(2 \times 10^{-2})\), after carrying out the multiplication, is \(6 \times 10^3\).

Key Concepts

Multiplying Powers of 10

When you encounter multiplication in scientific notation involving powers of 10, a key step is to address these powers separately. Scientific notation simplifies large and small numbers by expressing them as a product of a number called a coefficient and a power of 10. When multiplying powers of 10, the process is straightforward. You simply add the exponents. For example, consider the expression \(10^5 \times 10^{-2}\). Both numbers share the same base, which is 10.

• The base in both terms is the same (10).
• Add the exponents: \(5 + (-2) = 3\).
• Result: \(10^3\).

This rule makes calculations easy and reduces the complexity of dealing with numbers that are very large or very small.

Exponent Properties

Understanding exponent properties is critical when working with powers of 10. The properties of exponents provide a handy set of rules to navigate these calculations efficiently. The primary property used when multiplying is the addition of exponents. Here’s a quick summary:

• Product of Powers Property: For any base \(a\), \(a^m \times a^n = a^{m+n}\).
• This means you add the exponents together when multiplying terms with the same base.
• This property holds true regardless of whether the exponents are positive, negative, or zero.

Being familiar with these properties helps you manipulate and simplify large expressions with ease, making you more agile in solving a variety of problems involving powers.

Coefficients Multiplication

When dealing with scientific notation, another important piece of the puzzle is multiplying the coefficients. These coefficients are the numerical parts of the scientific notation and need to be calculated separately from the powers of 10. By focusing on the coefficients first, you simplify the expression.Consider the example: \((3 \times 10^5)(2 \times 10^{-2})\). Here’s how you approach multiplying the coefficients:

• Identify the coefficients that you are working with: 3 and 2.
• Multiply the coefficients: \(3 \times 2 = 6\).

Once you have the product of the coefficients, you can then easily pair this result with the product of the powers of 10, which is calculated separately. This organized method ensures accuracy and clarity in your computations.