Problem 93 - Algebra and Trigonometry

Question

\(91-96\) . Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$ \frac{1.295643 \times 10^{9}}{\left(3.610 \times 10^{-17}\right)\left(2.511 \times 10^{6}\right)} $$

Step-by-Step Solution

Answer

The answer is \(1.428 \times 10^{19}\), rounded to 4 significant figures.

Step 1: Understand the Problem

We need to divide the number \(1.295643 \times 10^{9}\) by the product of \(3.610 \times 10^{-17}\) and \(2.511 \times 10^{6}\). This involves using scientific notation and applying the Laws of Exponents.

Step 2: Multiply the Denominator

First, calculate the product of \(3.610 \times 10^{-17}\) and \(2.511 \times 10^{6}\): \[3.610 \times 2.511 = 9.06681\]For the exponents: \[10^{-17} \times 10^{6} = 10^{-17 + 6} = 10^{-11}\]This gives us a denominator of \(9.06681 \times 10^{-11}\).

Step 3: Divide the Numbers

Now divide \(1.295643 \times 10^{9}\) by \(9.06681 \times 10^{-11}\). Use the following rules for division in scientific notation:For the coefficients:\[\frac{1.295643}{9.06681} \approx 0.142845 \]For the exponents:\[\frac{10^{9}}{10^{-11}} = 10^{9 - (-11)} = 10^{20}\]This results in: \[0.142845 \times 10^{20}\].

Step 4: Convert to Proper Scientific Notation

To express \(0.142845 \times 10^{20}\) in proper scientific notation, move the decimal point one place to the right: \[1.42845 \times 10^{19}\].

Step 5: Round Off to Significant Figures

The initial data \(3.610 \times 10^{-17}\) has 4 significant figures, and the rest of the numbers have 6. Thus, we round the final answer to 4 significant figures: \[1.428 \times 10^{19}\].

Key Concepts

Understanding the Laws of Exponents

In the exercise given, we had to deal with numbers expressed in scientific notation, meaning they were written in the form of a coefficient times 10 raised to an exponent. When working with such numbers, the laws of exponents become very useful. These laws allow us to simplify the multiplication and division of numbers in scientific notation without manually handling large or small numbers.

• When multiplying numbers in scientific notation, you add their exponents. For example, the product of numbers like \(10^a \times 10^b = 10^{a+b}\).
• When dividing, you subtract the exponent of the denominator from the exponent of the numerator, as in \(\frac{10^c}{10^d} = 10^{c-d}\).
• These simple rules make complex calculations more manageable and save time.

Understanding these laws is vital for handling scientific notation efficiently. When faced with multiplying or dividing, always remember to act on the coefficients and exponents separately.

Rounding to Significant Figures

Significant figures are the digits in a number that contribute to its precision. This includes all non-zero digits, any zeros between significant figures, and trailing zeros in decimal parts. In calculations, it is crucial to round results to the correct number of significant figures to reflect the precision of the given data.When performing operations:

• The result should be rounded to the least number of significant figures present in any of the numbers involved in the operation.
• In our example, the number \(3.610 \times 10^{-17}\) has 4 significant figures, which limits our final answer's precision to 4 significant figures as well.
• Rounding affects the accuracy of your results, and respecting the significant figures ensures the data's integrity is maintained.

Keeping significant figures in mind helps in understanding how precise calculated results are based on the precision of input data.

Using a Calculator for Scientific Notation

Handling scientific notation and performing calculations manually can be prone to error, especially with complex numbers. This is where calculators become indispensable tools. Most scientific calculators facilitate operations with scientific notation directly.Here's how to use them effectively:

• Look for the "EXP" or "EE" button on your calculator. This button is used to enter numbers in scientific notation easily.
• For example, to enter \(1.295643 \times 10^9\), type "1.295643," press "EXP/EE," then type "9." This sequence allows accurate inputs of numbers in scientific notation.
• When performing calculations such as multiplication or division, first enter the complete number with scientific notation, then use appropriate arithmetic keys.
• After reaching a result, check for correctness using the significant figures and ensuring it adheres to proper scientific notation.

Using the calculator efficiently helps to save time and prevents errors during calculations. Understanding these features ensures that even the most challenging parts of numeric operations are manageable.