Problem 63 - Elementary and Intermediate Algebra

Question

Use scientific notation to perform the calculations. Give all answers in scientific notation and standard notation. \(\left(8.4 \times 10^{-13}\right)\left(4.8 \times 10^{9}\right)\)

Step-by-Step Solution

Answer

The product is \(4.032 \times 10^{-3}\) in scientific notation and 0.004032 in standard notation.

Step 1: Multiply the Base Numbers

The first step is to multiply the base numbers of the scientific notation expressions. This means we multiply 8.4 by 4.8. Calculate: \[8.4 \times 4.8 = 40.32\]

Step 2: Add the Exponents

In scientific notation, when you multiply numbers, you add their exponents. Here, we need to add the exponents of 10: -13 and 9. Calculate:\[-13 + 9 = -4\]

Step 3: Write in Scientific Notation

Now that we have both the base number and the new exponent, we can write the result in scientific notation. Combine the results from Steps 1 and 2:\[40.32 \times 10^{-4}\]

Step 4: Express in Proper Scientific Notation

In scientific notation, the base number should be between 1 and 10. Adjust 40.32 to 4.032 by moving the decimal one place to the left, which means increasing the exponent by 1:\[4.032 \times 10^{-3}\]

Step 5: Convert to Standard Notation

Convert the number from scientific notation to standard notation by moving the decimal point three places to the left, because the exponent is -3:\[4.032 \times 10^{-3} = 0.004032\]

Key Concepts

Multiplying in Scientific Notation

When multiplying numbers in scientific notation, it's essential to handle the base numbers and the exponents correctly. First, multiply the base numbers, which are the numbers before the exponent bases of 10. In the example, you multiply 8.4 by 4.8.

• This calculation gives you 40.32.

Next, deal with the exponents. In scientific notation, you add the exponents of 10 when multiplying. For instance, if you have \(10^{-13}\) and \(10^{9}\), you add -13 and 9, resulting in -4.

• Adding the exponents simplifies the problem.

This process isolates these two operations and prepares the problem for reassembly in its scientific notation form.

Exponents in Scientific Notation

Working with exponents is a key part of scientific notation. In multiplication, the exponents tell you how to scale the result by determining the placement of the decimal. Once you've handled the base multiplication, the next task is exponent arithmetic.

• For \(10^{-13}\) and \(10^{9}\), you add the exponents: -13 + 9, which equals -4.
• This new exponent, -4, combines with your base result.

Understanding this step is crucial, as it maintains the integrity of the scientific notation while simplifying the multiplication process. Remember, adding exponents only happens during multiplication, unlike subtraction or division, which has different rules.

Conversion to Standard Notation

After completing the multiplication in scientific notation, you may need to express the answer in standard notation. To do this, you adjust the decimal placement according to the exponent result.

• For example, if you end with \(4.032 \times 10^{-3}\), the exponent -3 tells you to move the decimal three places to the left.
• This converts \(4.032 \times 10^{-3}\) into 0.004032 in standard notation.

This process might seem complex at first, but with practice, it becomes intuitive. Always keep in mind how the exponent guides the decimal movement—left for negative exponents and right for positive ones. This skill effectively transitions numbers between scientific and standard notation.