Problem 47 - Algebra 1

Question

CALCULATOR Use a calculator to evaluate the expression. Write the result in scientific notation and in decimal form. $$ 2,000,000 \cdot 12,000 $$

Step-by-Step Solution

Answer

The result in scientific notation is \(2.4 \times 10^{10}\) and in decimal form is \(24,000,000,000\).

Step 1: Convert each number to scientific notation

First, convert \(2,000,000\) and \(12,000\) into scientific notation. \(2,000,000\) can be written as \(2 \times 10^6\) and \(12,000\) as \(1.2 \times 10^4\).

Step 2: Multiply the base numbers and the powers

Now, multiply the base numbers (2 and 1.2) and then add the exponents of ten together (6 and 4). This gives \(2.4 \times 10^{10}\).

Step 3: Convert the scientific notation to decimal form

Finally, convert the scientific notation to decimal form. This gives \(24,000,000,000\).

Key Concepts

Understanding Multiplication in Scientific Notation

When we are faced with large numbers, multiplication can seem intimidating. However, breaking it down using scientific notation makes it simpler. Scientific notation forms allow us to work with smaller, more manageable numbers. Take, for example, our problem of multiplying 2,000,000 by 12,000.

First, express each number in scientific notation:

• 2,000,000 is written as \(2 \times 10^6\)
• 12,000 is written as \(1.2 \times 10^4\)

Now, when multiplying numbers in scientific notation, follow two steps:

• Multiply the base numbers: \(2 \times 1.2 = 2.4\)
• Add the exponents: \(10^6 \times 10^4 = 10^{(6+4)} = 10^{10}\)

This results in the scientific notation: \(2.4 \times 10^{10}\), simplifying the process substantially.

Decimal Form and Its Calculation

After multiplying in scientific notation, it's useful to convert the result back to decimal form. This helps understand the actual value of the product more concretely. The scientific result we obtained was \(2.4 \times 10^{10}\).

To convert this to decimal form, follow these steps:

• The number \(2.4\) stays as is, forming the base of the decimal.
• The exponent \(10^{10}\) tells you how many places the decimal point should move to the right. Here, move it 10 places right.

This process converts \(2.4 \times 10^{10}\) to a decimal: 24,000,000,000. It's crucial to understand this conversion as it expresses the grandeur of the number in an everyday format.

Calculator Usage For Large Numbers

Calculators are invaluable for dealing with large numbers, especially when multiplying in scientific notation. To tackle our initial problem using a calculator involves a few key steps.

First, enter each number into the calculator using its scientific notation:

• Input \(2\) and use the exponent function to enter \(10^6\).
• Next, input \(1.2\) and use the exponent function to enter \(10^4\).

Computers like calculators will automatically manage the order of operations, efficiently handling the multiplication of bases and the addition of exponents.

Make sure to review your calculator's manual if you are unfamiliar with entering scientific notation, as different calculators may have varied steps. Ultimately, these tools provide accurate results quickly and are excellent aids for homework.