Problem 88 - Elementary Algebra

Question

Find the product for the following problems. Write the result in scientific notation. $$ \left(7.3 \times 10^{4}\right)\left(2.1 \times 10^{-8}\right) $$

Step-by-Step Solution

Answer

Answer: The product is 1.533 × 10^-3.

Step 1: Multiply the coefficients

We will first multiply the coefficients (7.3 and 2.1): $$ 7.3 \times 2.1 = 15.33 $$

Step 2: Add the exponents

Now, we will add the exponents (4 and -8): $$ 4 + (-8) = -4 $$

Step 3: Rewrite the result in scientific notation

Finally, we will rewrite our result (15.33) with the new exponent (-4) in scientific notation: $$ 15.33 \times 10^{-4} = 1.533 \times 10^{1} \times 10^{-4} = 1.533 \times 10^{-3} $$ So, the product of the given numbers is: $$ \left(7.3 \times 10^{4}\right)\left(2.1 \times 10^{-8}\right) = 1.533 \times 10^{-3} $$

Key Concepts

Multiplying Coefficients

When dealing with numbers in scientific notation, the first step often involves multiplying the coefficients. A coefficient is the number that appears in front of the base 10 with its exponent. In our example, the coefficients are 7.3 and 2.1. When multiplying these coefficients, we simply use basic arithmetic:

• Multiply the numbers directly: \(7.3 \times 2.1 = 15.33\).

This product of 15.33 is not yet in the final scientific notation form because scientific notation typically represents a number as a coefficient between 1 and 10. We hold this result to adjust later as we finalize the entire mathematical expression in scientific notation.

Adding Exponents

Adding exponents is the next crucial step when you multiply numbers expressed in scientific notation. The important thing to remember here is that you only add the exponents of the powers of 10, not the coefficients. In our example, the exponents are 4 and -8. So,

• Add these exponents: \(4 + (-8) = -4\).

The result, -4, will serve as the new exponent for the base 10 in the scientific notation. This is because when you multiply powers of 10, you add their exponents according to the property \(a^m \times a^n = a^{m+n}\). So, the task is simplifying the expression while maintaining the validity of exponents.

Mathematical Notation

Mathematical notation allows us to communicate complex ideas in a concise form. Scientific notation is a part of this system, especially useful for expressing very large or very small numbers. Writing our previous result, 15.33, in scientific notation requires two adjustments:

• Adjust the coefficient to fall between 1 and 10. Here, that means rewriting 15.33 as 1.533 by moving the decimal one place left. This increases the overall exponent by 1 because of the movement.
• The final scientific notation form combines this new coefficient with the adjusted exponent: \(1.533 \times 10^{-3}\).

This notation provides a clear view of the scale of a number while maintaining accuracy, simplicity, and ease of use in calculations. It is a fundamental part of scientific, engineering, and mathematical equations.