Problem 94 - 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. $$ \frac{(73.1)\left(1.6341 \times 10^{28}\right)}{0.0000000019} $$

Step-by-Step Solution

Answer

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

Step 1: Convert Decimal to Scientific Notation

Convert the denominator, which is in decimal form, into scientific notation. The number is 0.0000000019. In scientific notation, this becomes \(1.9 \times 10^{-9}\).

Step 2: Rewrite the Expression

Substitute the scientific notation into the expression, rewriting it as: \[\frac{(73.1)(1.6341 \times 10^{28})}{1.9 \times 10^{-9}}.\]

Step 3: Calculate Numerator in Scientific Notation

Multiply \(73.1\) by \(1.6341 \times 10^{28}\). First multiply the coefficients: \(73.1 \times 1.6341 = 119.42621\)So, the numerator in scientific notation becomes:\(1.1942621 \times 10^{30}\).

Step 4: Apply Division of Scientific Notation

Divide the number in the numerator by the number in the denominator using the laws of exponents. \[\frac{1.1942621 \times 10^{30}}{1.9 \times 10^{-9}}\]To simplify, divide the coefficients and subtract the exponents:\(\frac{1.1942621}{1.9} = 0.628559\)Then compute the powers of 10:\(10^{30 - (-9)} = 10^{39}\).The result is:\(0.628559 \times 10^{39}\).

Step 5: Convert Result to Proper Scientific Notation

Convert \(0.628559 \times 10^{39}\) to proper scientific notation. Move the decimal point to get:\(6.28559 \times 10^{38}\).

Step 6: Round to Appropriate Significant Digits

The least number of significant figures in the original data is 2 (from the number 1.9). Thus, round \(6.28559 \times 10^{38}\) to two significant figures:\(6.3 \times 10^{38}\).

Key Concepts

Laws of Exponents

Understanding the Laws of Exponents is essential when working with scientific notation. These laws make it easier to multiply and divide exponential expressions by managing the powers of 10 efficiently. Here's a quick rundown:

• When multiplying exponential expressions with the same base, add the exponents:
  \(a^m \times a^n = a^{m+n}\)
• When dividing, subtract the exponents:
  \(\frac{a^m}{a^n} = a^{m-n}\)
• Raising a power to another power involves multiplying the exponents:
  \((a^m)^n = a^{m \times n}\)

In our exercise, we applied these principles during division. Specifically, we used the rule of subtracting the exponents to simplify powers of 10, transforming \(10^{30} \div 10^{-9}\) into \(10^{39}\). This simplification is crucial for obtaining correct and manageable results.

Significant Figures

Significant figures play an important role in both precision and communication of mathematical results. They tell us how accurate our measurements and calculations are supposed to be. Here are some key points:

• Non-zero digits are always significant.
• Any zeros between two significant numbers are significant.
• Leading zeros are not significant, but trailing zeros in a decimal number are significant.

In this task, we see three numbers: 73.1, 1.6341, and 1.9. The number with the least number of significant figures is 1.9, which has only two significant figures. Thus, our final answer, \(6.3 \times 10^{38}\), is rounded to match these significant figures according to the rules of significant figures in calculations. This ensures that our result does not suggest more accuracy than the initial data allowed.

Mathematical Operations

When performing mathematical operations like multiplication and division in scientific notation, it is crucial to handle both the coefficients and exponents systematically:

• Multiply or divide the coefficients (these are the numbers in front of the powers of 10).
• Apply the laws of exponents to manage the powers of 10 accordingly.

For example, we first multiplied the coefficients \(73.1\) and \(1.6341\), resulting in \(119.42621\). Then we adjusted the final expression using the laws of exponents. Following these steps systematically assures that each part of the expression is accurately computed, leading to reliable results even in complex equations.

Multiplication and Division in Scientific Notation

Multiplying and dividing numbers in scientific notation involves a straightforward process that emphasizes managing coefficients and exponents:

• For multiplication, like \((73.1)(1.6341 \times 10^{28})\), multiply the coefficients:
  \(73.1 \times 1.6341 = 119.42621\), then sum the exponents:
  \(10^0 \times 10^{28} = 10^{28}\).
• For division, such as \(\frac{1.1942621 \times 10^{30}}{1.9 \times 10^{-9}}\), divide the coefficients:
  \(\frac{1.1942621}{1.9} = 0.628559\), and subtract the exponents:
  \(10^{30 - (-9)} = 10^{39}\).

This structured methodology simplifies otherwise complex calculations, as shown in the original problem's solution. The resulting expression is a neatly organized scientific notation, which can then be rounded based on significant figures as required.