Problem 91 - 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. $$ \left(7.2 \times 10^{-9}\right)\left(1.806 \times 10^{-12}\right) $$

Step-by-Step Solution

Answer

The answer is \(1.3 \times 10^{-20}\).

Step 1: Apply the Commutative Law of Multiplication

First, we rearrange the expression \( \left(7.2 \times 10^{-9}\right)\left(1.806 \times 10^{-12}\right) \) into two separate products: the product of the coefficients \(7.2\) and \(1.806\), and the product of the powers of ten \(10^{-9} \cdot 10^{-12}\).

Step 2: Multiply the Coefficients

Calculate the product of the coefficients. Multiply \(7.2\) by \(1.806\) using a calculator:\[7.2 \times 1.806 = 12.9792\]

Step 3: Apply the Laws of Exponents

Use the Laws of Exponents to multiply \(10^{-9}\) and \(10^{-12}\). Add the exponents:\[10^{-9} \times 10^{-12} = 10^{-9-12} = 10^{-21}\]

Step 4: Combine Results

Combine the results from Step 2 and Step 3. The expression becomes:\[12.9792 \times 10^{-21}\]

Step 5: Convert to Scientific Notation

Adjust the coefficient \(12.9792\) to be between 1 and 10. We move the decimal one place to the left:\[12.9792 = 1.29792 \times 10^1\]Multiply this by \(10^{-21}\):\[1.29792 \times 10^1 \times 10^{-21} = 1.29792 \times 10^{-20}\]

Step 6: Consider Significant Digits

The original data \(7.2\) and \(1.806\) have two and four significant digits respectively. Therefore, we round the final result to two significant digits:\[1.29792 \approx 1.3\]

Step 7: Write Final Answer

Combine the significant digits and the power of ten to express the final answer:\[1.3 \times 10^{-20}\]

Key Concepts

Laws of Exponents

When dealing with numbers in scientific notation, understanding the laws of exponents is key to simplifying expressions effectively. These principles are especially useful to manage large or small numerical values, often represented in scientific notation. At its core, the laws of exponents simplify operations involving powers. Here are a few basic rules to remember:

• Multiplying like bases: Add the exponents together. For example, with the expression \(a^m \times a^n = a^{m+n}\).
• Dividing like bases: Subtract the exponents. This is represented as \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a power: Multiply the exponents. For instance, \((a^m)^n = a^{m \times n}\).

By applying these rules, you can simplify expressions involving exponents to a single power, which is essential in tasks like multiplying scientific notations. For example, in our multiplication problem, multiplying \(10^{-9}\) and \(10^{-12}\) results in \(10^{-21}\), achieved by simply adding the exponents \(-9\) and \(-12\). These rules make it easier to work across various mathematical and scientific calculations.

Significant Digits

Significant digits play a crucial role in precision and accuracy when working with numbers, particularly in scientific calculations. They represent the digits in a number that contribute to its measurement's accuracy. Here's why they are important:

• They highlight which numbers are reliable and meaningful in your calculation.
• Rounding to significant digits ensures that your final result is not more precise than the measurements used to obtain it.

In the problem at hand, the numbers \(7.2\) has two significant digits, while \(1.806\) has four. Therefore, when we perform the operations and obtain a number like \(1.29792\), it should be rounded to match the fewest number of significant digits from the original data—in this case, two, making the answer \(1.3 \times 10^{-20}\). This practice helps to maintain consistency in the precision of values presented.

Multiplication of Scientific Notation

Multiplying numbers in scientific notation can simplify dealing with very large or small numbers, which is common in scientific calculations. The process involves two main steps:

• First, multiply the coefficients (the numerical parts) of the scientific notations as in regular multiplication.
• Second, use the laws of exponents to combine the powers of ten.

In our mathematical operation, consider \((7.2 \times 10^{-9}) \times (1.806 \times 10^{-12})\). Here's a brief rundown of the steps:1. Multiply the coefficients: \(7.2 \times 1.806 = 12.9792\).2. Combine the exponents using addition: \(10^{-9} \times 10^{-12} = 10^{-21}\).3. Finally, adjust the result to proper scientific notation by ensuring that the coefficient is between 1 and 10. In our case, \(12.9792\) becomes \(1.29792 \times 10^1\), resulting in the expression \(1.29792 \times 10^{-20}\).Simplifying this gives the final answer, \(1.3 \times 10^{-20}\), after adjusting for significant digits.