Problem 91 - College Algebra

Question

Perform the indicated computations. Write the answers in scientifi c notation. If necessary, round the decimal factor in your scientifi c notation answer to two decimal places. $$\left(6.1 \times 10^{-8}\right)\left(2 \times 10^{-4}\right)$$

Step-by-Step Solution

Answer

The result of the multiplication \((6.1 \times 10^{-8})(2 \times 10^{-4})\) in scientific notation is \(1.22 \times 10^{-11}\).

Step 1: Identify numbers in scientific notation

First, identify the numbers presented in scientific notation, which are \(6.1 \times 10^{-8}\) and \(2 \times 10^{-4}\)

Step 2: Multiply the decimal parts

Then, multiply the two decimal parts of the numbers together. In this case, multiply 6.1 and 2. The result is 12.2.

Step 3: Add the exponents of 10

Next, add the exponents of each 10 component together. So, -8 and -4 will become -12.

Step 4: Write answer in scientific notation

Finally, write the answer in scientific notation. Here, it's necessary to adjust the decimal part to have only one digit before the decimal point, so 12.2 becomes 1.22 and the exponent increases by one, thus \(1.22 \times 10^{-11}\)

Key Concepts

Multiplication of Exponents

When dealing with numbers in scientific notation, one of the key operations is the multiplication of exponents. In scientific notation, numbers are expressed as a product of a decimal and a power of 10, like this: \(a \times 10^n\). To multiply such numbers, you multiply their decimal parts and add their exponents.
Imagine you are given two numbers: \(6.1 \times 10^{-8}\) and \(2 \times 10^{-4}\). We start by focusing on the powers of 10. As each number can be seen as \(10\) raised to a power, combining them involves adding exponents:

• The exponent from the first number is \(-8\).
• The exponent from the second number is \(-4\).
• Therefore, the sum of exponents is \(-8 + (-4) = -12\).

This operation is straightforward, yet fundamental to working with exponential expressions in scientific notation, turning complex computations into simple addition.

Decimal Adjustment

After multiplying decimals, it's essential to ensure the proper format of scientific notation. Scientific notation requires the decimal part, also known as the mantissa, to have exactly one digit before the decimal point.
For our example, multiplying the decimals \(6.1\) and \(2\) results in \(12.2\). However, since scientific notation demands only one digit before the decimal, we must adjust \(12.2\) to \(1.22\). This process is called decimal adjustment.

• Shift the decimal one place to the left to get \(1.22\).
• To maintain the value of the number, increase the exponent by one, taking into account that we moved the decimal point leftwards.

After adjustment, the expression becomes \(1.22 \times 10^{-11}\).
Remember, this adjustment is crucial for correct scientific notation, ensuring consistency and clarity.

Multiplying Decimals

Multiplying decimals is sometimes perceived as tricky, but it's manageable with care and practice. Here, we multiply the decimal parts of the numbers given in scientific notation.
Taking \(6.1\) and \(2\) as our decimals, follow these steps to multiply:

• Align the numbers by placing them for multiplication while ignoring the decimal place initially.
• Multiply as if they were whole numbers. \(6.1\) times \(2\) delivers \(12.2\).
• Place the decimal in the result, ensuring accuracy by counting the total decimal places in the original numbers.

In this example, you maintain the precision of \(12.2\). Remember, mastering decimal multiplication enhances your skill with scientific notation and makes handling real-world numbers more straightforward.

Exponents Addition

Adding exponents is fundamental in the realm of scientific notation. When you multiply numbers like \(6.1 \times 10^{-8}\) with \(2 \times 10^{-4}\), their decimal parts and their exponents on the 10s are addressed separately.
Once you focus on the multiplication of decimal parts (as previously discussed), turn your attention to the operation of their exponents:

• Consider \(10^{-8}\) and \(10^{-4}\) separately.
• Add the exponents together: \(-8\) plus \(-4\) equals \(-12\).

This sums up the product of the powers of 10, finding a new base-10 exponent as \(10^{-12}\). Moreover, this simplification allows calculations with large or small numbers to remain efficient and manageable.
Understanding and applying the concept of exponents addition ensures success in any task requiring scientific notation.