Problem 90 - College Algebra

Question

Write each number in scientific notation and use scientific notation to perform the operation(s). Express the answer in scientific notation. $$ \frac{282,000,000,000}{0.00141} $$

Step-by-Step Solution

Answer

The solution to \(\frac{282,000,000,000}{0.00141}\) in scientific notation is \(2.00 \times 10^{14}\).

Step 1: Conversion to Scientific Notation

First, convert the numbers into scientific notation. The number 282,000,000,000 can be written as \(2.82 \times 10^{11}\) and 0.00141 becomes \(1.41 \times 10^{-3}\).

Step 2: Perform Division Operation

The operation becomes \( \frac{2.82 \times 10^{11}}{1.41 \times 10^{-3}}\). Now, to process this, perform the division operation on the decimal parts and on the powers of 10 separately. Use the laws of exponents to subtract the exponent in the denominator from the exponent in the numerator.

Step 3: Implement Division Operation

\(\frac{2.82}{1.41} = 2.00\) and \(10^{11} \div 10^{-3} = 10^{11 + 3} = 10^{14}\). So, the result becomes \(2.00 \times 10^{14}\).

Key Concepts

Converting to Scientific Notation

When dealing with very large or very small numbers, converting them to scientific notation makes calculations like division much more manageable. The process is straightforward: for a large number, count how many places you move the decimal point to the left until you have a number between 1 and 10. For a small number, you move the decimal to the right. This count becomes the exponent of 10, with large numbers getting a positive exponent and small numbers getting a negative exponent.

For example, with the number 282,000,000,000, we move the decimal 11 places to the left to get 2.82. This gives us the scientific notation form of \(2.82 \times 10^{11}\). Similarly, for 0.00141, moving the decimal point three places to the right gives us 1.41, so its scientific notation is \(1.41 \times 10^{-3}\). It's essential to ensure the coefficient (the number in front of \(10\)) is indeed between 1 and 10; otherwise, it's not properly in scientific notation.

Laws of Exponents

The laws of exponents are fundamental rules that make working with exponential expressions simpler. Two particularly useful laws for division are the quotient of powers rule and the power of a quotient rule.

When dividing exponents with the same base, like \(10^{n} \div 10^{m}\), you subtract the exponents (\

Performing Division Operation

To correctly execute division of numbers in scientific notation, separate the operation into two parts: division of the coefficients (the numbers in front of the \(10\)) and division of the powers of ten. Start by dividing the coefficients normally. For the example given, \(2.82 \div 1.41\), you'll get 2.00.

Next, address the powers of ten. Apply the rules of exponents by subtracting the exponent in the denominator from the exponent in the numerator \(10^{11} \div 10^{-3} = 10^{11} + ( - (-3)) = 10^{14}\). This process simplifies the division operation, resulting in a final answer in scientific notation: \(2.00 \times 10^{14}\). Remember that precision matters when dividing the coefficients, and any changes in the coefficient affect the overall result. By handling the operations separately and accurately, you'll consolidate your understanding of scientific notation and division with exponents.