Problem 95 - 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. $$ \frac{(0.0000162)(0.01582)}{(594,621,000)(0.0058)} $$

Step-by-Step Solution

Answer

The result is \( 7.4 \times 10^{-14} \).

Step 1: Convert Numbers to Scientific Notation

Convert all numbers into scientific notation. The numbers are given as \( 0.0000162 \), \( 0.01582 \), \( 594,621,000 \), and \( 0.0058 \).- \( 0.0000162 = 1.62 \times 10^{-5} \)- \( 0.01582 = 1.582 \times 10^{-2} \)- \( 594,621,000 = 5.94621 \times 10^8 \)- \( 0.0058 = 5.8 \times 10^{-3} \)

Step 2: Set Up the Expression Using Scientific Notation

Substitute the scientific notation into the given expression:\[ \frac{(1.62 \times 10^{-5})(1.582 \times 10^{-2})}{(5.94621 \times 10^8)(5.8 \times 10^{-3})} \]

Step 3: Apply the Multiplication Laws of Exponents

Multiply the numerators and the denominators separately, using the Laws of Exponents:- Numerator: \((1.62 \times 1.582) \times (10^{-5} \times 10^{-2}) = 2.56324 \times 10^{-7}\)- Denominator: \((5.94621 \times 5.8) \times (10^8 \times 10^{-3}) = 34.487018 \times 10^{5}\)

Step 4: Divide the Terms

Divide the coefficients and subtract the exponents:\[ \frac{2.56324}{34.487018} \times 10^{-7-5} = 0.0743 \times 10^{-12} \]

Step 5: Convert Back to Scientific Notation

Adjust the scientific notation:\( 0.0743 \times 10^{-12} \) becomes \( 7.43 \times 10^{-14} \)

Step 6: Determine Significant Figures

The number with the least significant figures from data is \( 0.0058 \) (2 significant figures), so round the final answer to 2 significant figures:\( 7.4 \times 10^{-14} \)

Key Concepts

Laws of Exponents

When working with scientific notation, the Laws of Exponents simplify the process of multiplication and division. They help us handle expressions with powers more easily. Let's break down the rules used in our problem:

• Multiplication: When multiplying numbers in scientific notation, multiply the base numbers and add their exponents.

For example, \[ (a \times 10^m) \times (b \times 10^n) = (a \cdot b) \times 10^{m+n} \]In the exercise, we multiply the bases 1.62 and 1.582 to get 2.56324. Similarly, we sum the exponents \(-5\) and \(-2\) to get \(-7\).

• Division: For division, you divide the base numbers and subtract the exponents.

For example, \[ \frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n} \]In our problem, the numerator is 2.56324, and the denominator is 34.487018. By dividing these, and subtracting the exponent \(-7\) from \(5\), you end up with \(10^{-12}\). Using these laws allows us to simplify complex equations with ease.

Significant Figures

Understanding significant figures is crucial in scientific calculations to ensure precision. They are the digits in a number that contribute to its accuracy. Here's how significant figures play a role:

• Identify Significant Figures: Check each number involved in your calculation for its significant figures.

In our exercise, \(0.0058\) has the least significant figures (2). Thus, our final result should also reflect this level of precision.

• Rounding: After performing calculations, round your answer to match the original data with the fewest significant figures.

Finally, our calculated answer \(7.43 \times 10^{-14}\) is rounded to two significant figures, becoming \(7.4 \times 10^{-14}\).This practice ensures that your results are not overly precise compared to your measurement data.

Numerator and Denominator

In division problems, especially with fractions, understanding the roles of the numerator and denominator is essential. Here's how each part functions:

• Numerator: The numerator is the top part of a fraction. It represents the number being divided.

In our expression, \( (1.62 \times 10^{-5})(1.582 \times 10^{-2}) \), the numerator combines these two values. The result from multiplying them in scientific notation is \(2.56324 \times 10^{-7}\).

• Denominator: The denominator is the bottom part of a fraction, indicating what the numerator is being divided by.

In the given problem, \( (5.94621 \times 10^8)(5.8 \times 10^{-3}) \) forms the denominator. The product from this calculation is \( 34.487018 \times 10^{5} \).By dividing the numerator by the denominator, we get the simplified result. Recognizing these roles helps you effectively manage and solve fractional calculations, especially in scientific notation.