Problem 79 - Chemistry Matter and Change

Question

Complete the following multiplication and division problems in scientific notation. a. \(\left(4.8 \times 10^{5} \mathrm{km}\right) \times\left(2.0 \times 10^{3} \mathrm{km}\right)\) b. \(\left(3.33 \times 10^{-4} \mathrm{m}\right) \times\left(3.00 \times 10^{-5} \mathrm{m}\right)\) c. \(\left(1.2 \times 10^{6} \mathrm{m}\right) \times\left(1.5 \times 10^{-7} \mathrm{m}\right)\) d. \(\left(8.42 \times 10^{8} \mathrm{kL}\right) \div\left(4.21 \times 10^{3} \mathrm{kL}\right)\) e. \(\left(8.4 \times 10^{6} \mathrm{L}\right) \div\left(2.4 \times 10^{-3} \mathrm{L}\right)\) f. \(\left(3.3 \times 10^{-4} \mathrm{mL}\right) \div\left(1.1 \times 10^{-6} \mathrm{mL}\right)\)

Step-by-Step Solution

Answer

a) \(9.6 \times 10^8\) km², b) \(9.99 \times 10^{-9}\) m², c) \(1.8 \times 10^{-1}\) m², d) \(2.00 \times 10^5\) kL, e) \(3.5 \times 10^9\) L, f) \(3.0 \times 10^2\) mL.

Step 1: Multiplying Numbers

To solve part (a) \( (4.8 \times 10^5) \times (2.0 \times 10^3) \)First, multiply the decimal numbers:\(4.8 \times 2.0 = 9.6\)

Step 2: Adding Exponents in Multiplication

Next, add the exponents of 10 when multiplying: \(5 + 3 = 8\)So, part (a) becomes: \(9.6 \times 10^8 \text{ km}^2\)

Step 3: Multiplying Decimal Parts for Part b

For part (b), multiply the decimal parts:\(3.33 \times 3.00 = 9.99\)

Step 4: Adding Exponents for Part b

Add the exponents in part (b):\(-4 + (-5) = -9\)So, part (b) is:\(9.99 \times 10^{-9} \text{ m}^2\)

Step 5: Multiplying Decimal Parts for Part c

Multiply the decimal parts of the numbers in part (c):\(1.2 \times 1.5 = 1.8\)

Step 6: Adding Exponents for Part c

Add the exponents for part (c):\(6 + (-7) = -1\)So, part (c) is:\(1.8 \times 10^{-1} \text{ m}^2\)

Step 7: Dividing Decimal Parts for Part d

For division in part (d):\(8.42 \div 4.21 = 2.00\)

Step 8: Subtracting Exponents in Division

Subtract the exponents for division in part (d):\(8 - 3 = 5\)So, part (d) is:\(2.00 \times 10^5 \text{ kL}\)

Step 9: Dividing Decimal Parts for Part e

Divide the decimal numbers in part (e):\(8.4 \div 2.4 = 3.5\)

Step 10: Subtracting Exponents for Part e

Subtract the exponents for part (e):\(6 - (-3) = 9\)So, part (e) is:\(3.5 \times 10^9 \text{ L}\)

Step 11: Dividing Decimal Parts for Part f

Divide the decimal numbers in part (f):\(3.3 \div 1.1 = 3.0\)

Step 12: Subtracting Exponents for Part f

Subtract exponents for part (f):\(-4 - (-6) = 2\)So, part (f) is:\(3.0 \times 10^2 \text{ mL}\)

Key Concepts

Multiplication and Division in Scientific Notation

When dealing with large or small numbers, scientists often use scientific notation to simplify calculations. It involves expressing numbers as the product of a decimal and a power of ten, which makes multiplication and division easier. To multiply two numbers in scientific notation:

• Multiply the decimal parts.
• Add the exponents of the power of ten.

For example, in \[(4.8 \times 10^5) \times (2.0 \times 10^3)\]first multiply the decimal values: \[4.8 \times 2.0 = 9.6\]Then, add the exponents: \[5 + 3 = 8\]resulting in: \[9.6 \times 10^8\] kilometers squared.
For division, divide the decimal parts and subtract the exponents. For instance, \[(8.42 \times 10^8) \div (4.21 \times 10^3)\]requires:

• Divide the decimal values: \[8.42 \div 4.21 = 2.00\]
• Subtract the exponents: \[8 - 3 = 5\]

This gives \[2.00 \times 10^5\] kiloliters.

Exponent Rules

Exponents play a crucial role in scientific notation, especially when performing mathematical operations like multiplication and division. There are a few key rules to remember:

• When multiplying similar bases, add their exponents: \( a^m \times a^n = a^{m+n} \).
• When dividing, subtract the exponents of the numerators and denominators: \( \frac{a^m}{a^n} = a^{m-n} \).
• Negative exponents indicate a reciprocal: \( a^{-n} = \frac{1}{a^n} \).

Applying these rules ensures accurate results when working with powers of ten. For example, in \[3.33 \times 10^{-4} \times 3.00 \times 10^{-5}\]you get:- Multiply the exponents: \[ (-4) + (-5) = -9 \], resulting in \[ 9.99 \times 10^{-9} \].

Significant Figures

Significant figures reflect the precision of a measurement or calculation. They include all notable digits in a number, which is crucial when handling scientific data. Here's how to determine significant figures:

• All non-zero digits are significant.
• Any zeros between significant digits are significant (e.g., \[ 1002 \]).
• Leading zeros are not significant (e.g., \[ 0.0032 \]).
• Trailing zeros are significant only if they are after a decimal point (e.g., \[ 5.00 \]).

When performing calculations:- The result should reflect the least number of significant figures from the inputs. For example, \[(3.3 \div 1.1) = 3.0\], maintains two significant figures as in the original numbers.

Metric Units

Metric units form an essential part of scientific notation and measurement, providing a universally understood system of measurement. Understanding metric prefixes helps in working with diverse units. Common prefixes include:

• Kilo (\[ k \]) for \[ 10^3 \] (e.g., kilometer).
• Milli (\[ m \]) for \[ 10^{-3} \] (e.g., millimeter).
• Micro (\[ \mu \]) for \[ 10^{-6} \] (e.g., microliter).

Proper use of metric units ensures consistency in scientific communication. An example is converting between units like kiloliters and liters in division calculations: \[(8.4 \times 10^6 \text{ L}) \div (2.4 \times 10^{-3} \text{ L})\]. By accurately managing units, we achieve results that are both correct and useful for further scientific inquiry.

Mathematical Operations

Mathematical operations like multiplication and division reappear across various scientific and real-world applications, frequently simplified using scientific notation. Here’s how to approach them effectively:

• Ensure that the operations follow the order: parentheses, exponents, multiplication and division (from left to right), addition and subtraction (PEMDAS/BODMAS).
• Accurately apply scientific notation rules for optimal results; this reduces complexity and error, especially in extensive calculations.
• Cross-check results to verify correctness, noting the consistency with significant figures and the appropriate use of units.

For instance, managing operations in \[1.2 \times 10^6 \times 1.5 \times 10^{-7}\]requires careful attention to both the magnitude and the units involved, leading to accurate and meaningful results in scientific analysis.