Problem 104 - College Algebra

Question

Perform the indicated computations. Write the answers in scientifi c notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $$\frac{282,000,000,000}{0.00141}$$

Step-by-Step Solution

Answer

The result of the division expressed in scientific notation to two decimal places is \(2.00 \times 10^{11}\)

Step 1: Perform the Division

Determine the result of the division \( \frac{282,000,000,000}{0.00141} \) which equals \(200 billion or 200,000,000,000\)

Step 2: Convert to Scientific Notation

Express the result from step 1 in scientific notation. To do this, move the decimal point until there is one number to the left of the decimal. Count the number of places moved, which will be the exponent on the base 10. \(200,000,000,000\) becomes \(2 \times 10^{11}\), because the decimal point was moved 11 places to the left.

Step 3: Round to Two Decimal Places

In this case, the coefficient in the scientific notation is already at two decimal places, so no further rounding is necessary. The final result in scientific notation is \(2.00 \times 10^{11}\)

Key Concepts

Division of large numbers

Dividing large numbers, like \( \frac{282,000,000,000}{0.00141} \), might seem daunting at first. But with a structured approach, it becomes much more manageable. Here's how you can tackle it:

• Identify the Numbers: The number being divided, called the dividend, is \(282,000,000,000\). The number you are dividing by, called the divisor, is \(0.00141\).
• Perform the Division: Start by treating the dividend and divisor as whole numbers by removing the decimal point from the divisor. Multiply both the dividend and the divisor by 100,000 (or move the decimal point 5 places to the right in the divisor), which gives you \(282,000,000,000\) and \(141\). Now perform the division of these new numbers. The calculation simplifies the problem but gives you the same result.
• Check the Division: It's essential to cross-verify your answers to avoid mistakes, especially with such large numbers.

By understanding this method, you can handle even bigger numbers with ease.

Rounding in scientific notation

Scientific notation often requires rounding decimals to a certain number of places. In this exercise, we worked with two decimal places. Rounding helps make numbers more concise without losing significant accuracy:

• Identify the Decimal: When converting \(200,000,000,000\) to scientific notation, the decimal coefficient is first \(2\).
• Apply Rounding Rules: First, make sure to retain only the desired number of significant figures after the decimal. Here, \(2\) naturally has two decimal places expressed as \(2.00\) when rounded to two decimal places.
• Precision in Notation: Rounding ensures that your scientific notation remains precise but straightforward. Ensure always to note if more precision is necessary for specific calculations.

This practice becomes easier with time and ensures you're working accurately.

Exponents and powers of ten

Scientific notation is all about exponents and powers of ten. It provides an elegant way to express very large numbers:

• Understanding Exponents: The exponent tells you how many times to multiply the number by ten. For instance, in \(2 \times 10^{11}\), the exponent \(11\) indicates that the base number \(2\) is multiplied by ten, eleven times.
• Moving the Decimal: When converting a number like \(200,000,000,000\) into scientific notation, the decimal has been moved 11 places to the left, corresponding to the power of ten. This process helps transform what would be an unwieldy number into a more manageable format.
• An Efficient System: Using powers of ten not only simplifies calculations but also makes it easier to notice the scale of changes in large or small numbers.

By mastering exponents and powers of ten, tackling scientific calculations becomes a breeze.