Problem 94 - 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. $$ \left(8.2 \times 10^{8}\right)\left(4.6 \times 10^{4}\right) $$

Step-by-Step Solution

Answer

The solution is \(3.772 \times 10^{13}\)

Step 1: Multiplication of Decimal Part

Start by multiplying the numbers on the left side, so \(8.2 \times 4.6 = 37.72\).

Step 2: Addition of Exponents

Next, because both numbers are in scientific notation and have the same base, simply add the exponents. In this case, \(8 + 4 = 12\). Thus the exponent part is \(10^{12}\).

Step 3: Combine the Result

Now we need to combine step 1 and step 2, so the initial answer is \(37.72 \times 10^{12}\)

Step 4: Adjust to Correct Scientific Notation

For the final correct scientific notation, the number before the decimal should be between 1 and 10. Notice that our initial answer does not meet this standard. Therefore, we convert \(37.72 \times 10^{12}\) to \(3.772 \times 10^{13}\). We do this by moving the decimal point one place to the left and increasing the exponent by 1.

Step 5: Rounding the Decimal Part

Lastly, round off the decimal factor in the scientific notation answer to two decimal places. Here, it's already in the correct form and there is no need for rounding as it already has two decimal places: 3.77

Key Concepts

Decimal Multiplication

Decimal multiplication is an important step when dealing with numbers in scientific notation. When you have two numbers to multiply, like in our exercise, begin with multiplying the decimal parts. For example, with

• 8.2 and
• 4.6

the computation involves multiplying just these two to make 37.72.
This process often involves basic multiplication skills that you might already be familiar with. Take care to line up the decimal points correctly.
Once computed, this result becomes part of the eventual scientific notation output. However, you may need to make additional adjustments to fit the rules of scientific notation as will be explained in the next sections.

Exponent Addition

When numbers are in scientific notation and share the same base, you simply add the exponents together. In our example,

• 8.2 is multiplied by \(10^8\),
• 4.6 is multiplied by \(10^4\).

To continue with scientific notation, add these exponents: \(8 + 4 = 12\).
This happens because the laws of exponents indicate that when you multiply like bases, you should add the exponents.
This concept lets you combine powers of ten easily, streamlining any multiplication process involving scientific notation.

Scientific Notation Rules

Scientific notation provides a way to express very large or very small numbers conveniently. Here are some key features:

• The number before the decimal point should be from 1 to below 10.
• The notation looks like this: \(a \times 10^n\), where \(1 \leq a < 10\).

In the exercise, after computing \(37.72 \times 10^{12}\), we adjust to fit the rule by converting to \(3.772 \times 10^{13}\).
This involved shifting the decimal point one place to the left and increasing the exponent by 1.
This small shift aligns the number with scientific notation rules, ensuring clarity and uniformity.

Rounding Decimals

When working with scientific notation, rounding the decimal factor is often necessary for precision and simplicity. In our exercise, the decimal factor is 3.772.
According to the instructions, it needs rounding to two decimal places, leading to 3.77.
Rounding involves checking the digit right after your desired decimal place. If 5 or more, round up the last desired digit; if less than 5, round down.
This process ensures that results are easy to read and align with conventional scientific notation standards without losing essential accuracy.