Problem 72 - Algebra and Trigonometry

Question

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer correct to the number of significant digits indicated by the given data. $$ \frac{(73.1)\left(1.6341 \times 10^{28}\right)}{0.0000000019} $$

Step-by-Step Solution

Answer

The answer is \( 6.3 \times 10^{38} \).

Step 1: Convert to Scientific Notation

First, express every number in scientific notation. The number 73.1 can be considered as \( 7.31 \times 10^1 \). The number 0.0000000019 can be expressed in scientific notation as \( 1.9 \times 10^{-9} \). Thus the expression becomes \[ \frac{(7.31 \times 10^1)(1.6341 \times 10^{28})}{1.9 \times 10^{-9}}. \]

Step 2: Apply the Laws of Exponents

Utilize the law that allows for multiplication of powers and division of powers: \((a \times 10^n)(b \times 10^m) = (a \cdot b) \times 10^{n+m}\) and \(\frac{(a \times 10^n)}{(b \times 10^m)} = \frac{a}{b} \times 10^{n-m}\). Therefore, rewrite the expression as \[ \frac{(7.31 \cdot 1.6341) \times 10^{1+28}}{1.9 \times 10^{-9}} = \frac{11.945871}{1.9} \times 10^{29} \times 10^9. \]

Step 3: Simplify the Expression

Calculate the numerical fraction: \( \frac{11.945871}{1.9} \approx 6.2873021 \). Now apply the laws of exponents to simplify the power of ten: \( 10^{29 + 9} = 10^{38} \). So, the expression becomes \( 6.2873021 \times 10^{38} \).

Step 4: Determine Correct Number of Significant Figures

The original numbers had these significant figures: 73.1 (3 digits), 1.6341 (5 digits), and 0.0000000019 (2 digits). The result should be presented with 2 significant digits because that's the smallest number among them. Round 6.2873021 to two significant digits, resulting in 6.3.

Step 5: Final Scientific Notation

Write the final answer in scientific notation considering the rounded number: \( 6.3 \times 10^{38} \).

Key Concepts

Laws of Exponents

Understanding the laws of exponents is essential when solving problems involving scientific notation. These laws help simplify complex expressions with exponential terms. Let's break down a couple of fundamental rules of exponents that are particularly useful:

• Product of Powers: When multiplying numbers with the same base, you add their exponents: \[(a^m \times a^n = a^{m+n}).\]
• Division of Powers: When dividing numbers with the same base, you subtract the exponents: \[\left(\frac{a^m}{a^n} = a^{m-n}\right).\]
• Power of a Power: When raising an exponential term to another exponent, you multiply the exponents: \[(a^m)^n = a^{m \times n}).\]

In the context of converting to scientific notation, these rules allow us to manage and simplify expressions where the base of 10 is involved. For example, in our problem, you multiply \[10^{1+28}\] and simplify \[10^{29+9}\] using these laws. This makes the problem more manageable.Exponential laws are crucial for efficiently working through such calculations, so practice applying these rules can make solving problems much more straightforward.

Significant Figures

Significant figures are vital in scientific calculations to determine how precise a number is. They reflect the precision of measurements or calculations and guide rounding practices. Here’s some insight on determining significant figures:

• All non-zero numbers are significant. For example, 73.1 has three significant figures.
• Zeros between significant digits are significant. E.g., 1.6341 has five significant figures.
• Leading zeros, or zeros before the first significant digit, are not significant. Therefore, 0.0000000019 only has two significant figures.
• In multiplication or division, the result should have the same number of significant figures as the number having the least significant figures.

For our problem, the output should be rounded to two significant figures because the number 0.0000000019 has only two. This ensures consistency and reliability of the result with the data’s precision. Thus, 6.2873021 becomes 6.3 when appropriately rounded for scientific notation purposes.

Calculator Usage

Using a calculator can significantly simplify solving problems with scientific notation, especially when dealing with multiplication or division of large (or very small) numbers. Here’s how to make the most of a calculator for such exercises:

• Scientific Notation Mode: Most scientific calculators have a mode specifically for entering and computing with scientific notation. Look for a mode labelled as "SCI" or similar.
• Entering Numbers: Use the "EXP" or "EE" button to enter numbers in scientific notation. For example, to input \[1.9 \times 10^{-9},\] you would type "1.9 EXP -9".
• Parentheses for Operations: Utilize parentheses to maintain proper order and grouping of operations, which is crucial for accurate calculations.
• Review and Adjust: After performing calculations, verify the result, particularly the exponent, for any errors before rounding off to the applicable significant figures.

With a calculator, it's crucial to double-check your input and understand how the specific calculator model handles the operations, so always refer to the manual if you're unsure. This ensures accuracy in computations and final results, which in turn, strengthens comprehension of the scientific notation process.