Problem 74 - 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. $$ \frac{\left(3.542 \times 10^{-6}\right)^{9}}{\left(5.05 \times 10^{4}\right)^{12}} $$

Step-by-Step Solution

Answer

The expression evaluates to \(2.113 \times 10^{-108}\).

Step 1: Write the Expression in Scientific Notation

The given expression is already written in scientific notation: \[ \frac{(3.542 \times 10^{-6})^{9}}{(5.05 \times 10^{4})^{12}} \]

Step 2: Apply the Power of a Power Rule

Use the power of a power rule \((a^m)^n = a^{m \cdot n}\) to simplify the exponents.For the numerator: \((3.542 \times 10^{-6})^{9} = 3.542^9 \times (10^{-6})^{9} = 3.542^9 \times 10^{-54}\). For the denominator: \((5.05 \times 10^{4})^{12} = 5.05^{12} \times (10^{4})^{12} = 5.05^{12} \times 10^{48}\).

Step 3: Simplify the Exponents in the Fraction

Now simplify the expression using the results from Step 2: \[ \frac{3.542^9 \times 10^{-54}}{5.05^{12} \times 10^{48}} = \frac{3.542^9}{5.05^{12}} \times 10^{-54-48} = \frac{3.542^9}{5.05^{12}} \times 10^{-102} \]

Step 4: Calculate Powers and Division

Use a calculator to find \(3.542^9\) and \(5.05^{12}\) and divide the results. \[3.542^9 \approx 20294700.4504\] \[5.05^{12} \approx 9607039604049.8\] Now divide: \[ = \frac{20294700.4504}{9607039604049.8} \approx 2.11298 \times 10^{-6}\]

Step 5: Adjust to Correct Significant Digits

The original numbers had four significant digits, so the result should also be in four significant digits. Adjust the value to: \[ 2.113 \times 10^{-6} \]

Step 6: Compile Final Answer

Combine all steps to state the final answer in scientific notation: \[ 2.113 \times 10^{-108} \]

Key Concepts

Laws of Exponents

The Laws of Exponents are fundamental rules that help simplify expressions involving powers or exponents. These laws dictate how to handle operations like multiplication, division, and powers of powers with exponential expressions. When working with exponents, you might encounter:

• Product of Powers: When multiplying similar bases, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: When dividing similar bases, you subtract the exponents: \(a^m / a^n = a^{m-n}\).
• Power of a Power: When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).

Applying these laws correctly allows conversion of complex scientific notation to simpler forms. In the original problem, understanding and applying these rules helps transform and solve the entire expression.

Significant Digits

Significant digits, also known as significant figures, are digits that carry meaningful information about the precision of a number. They include all the certain digits in a number, plus one uncertain or estimated digit. The concept is crucial in scientific calculations to convey the accuracy and reliability of results.

• Non-zero digits are always significant.
• Any zeros between significant digits are significant.
• Leading zeros are not significant.
• Trailing zeros in a decimal number are significant.

Suppose the original numbers in the problem have four significant digits. The result must be expressed with the same precision to ensure accuracy. This involves rounding calculations to preserve the number of significant digits.

Power of a Power Rule

The Power of a Power Rule is one of the essential laws of exponents. It provides a straightforward method to simplify expressions involving exponents raised to another exponent. The rule states:

• \((a^m)^n = a^{m \cdot n}\)

This rule enables you to manage operations with compounds exponents by multiplying them together. For example, in the given problem, the powers in both the numerator and the denominator were simplified by applying this rule.
Simplifying such expressions accurately involves recognizing the base number and appropriately multiplying its exponents to achieve the correct power.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They have the form \(a^x\), where \(a\) is the base and \(x\) is the exponent. These functions are widely used in real-world applications, including population growth, radioactive decay, and compound interest.

• Exponential Growth: Occurs when the rate of change increases exponentially over time.
• Exponential Decay: Describes a process where the quantity decreases at a rate proportional to its current value.

In the context of the problem, understanding exponential functions helps interpret the relationship and manipulation of numbers expressed in scientific notation. Exponential functions inherently obey the laws of exponents, making them a primary focus when simplifying and calculating large or small numbers in scientific notation.