Problem 96 - 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. $$ \frac{\left(3.542 \times 10^{-6}\right)^{9}}{\left(5.05 \times 10^{4}\right)^{12}} $$

Step-by-Step Solution

Answer

The result is approximately \(1.63 \times 10^{-102}\).

Step 1: Understand the Problem

The problem requires us to evaluate the expression \( \frac{(3.542 \times 10^{-6})^9}{(5.05 \times 10^4)^{12}} \) using scientific notation and the laws of exponents.

Step 2: Apply the Law of Exponents

According to the laws of exponents, \((a \times 10^b)^n = a^n \times 10^{bn}\). Apply this to both the numerator and the denominator.

Step 3: Calculate the Numerator

For the numerator, apply the laws of exponents: \((3.542 \times 10^{-6})^9 = 3.542^9 \times 10^{-54}\). Use a calculator to find \(3.542^9\).

Step 4: Calculate the Denominator

For the denominator, apply the laws of exponents: \((5.05 \times 10^{4})^{12} = 5.05^{12} \times 10^{48}\). Use a calculator to find \(5.05^{12}\).

Step 5: Divide the Results

Divide the calculated values of the numerator and the denominator, i.e., \( \frac{3.542^9 \times 10^{-54}}{5.05^{12} \times 10^{48}} = \frac{3.542^9}{5.05^{12}} \times 10^{-102}\).

Step 6: Calculate Final Result and Significant Figures

Use a calculator to find \(\frac{3.542^9}{5.05^{12}}\) and then apply the exponent rules for combining the powers of 10. Round your answer to the correct number of significant figures, which is 3, since 3.542 has 4 significant figures and 5.05 has 3.

Key Concepts

Laws of Exponents

The laws of exponents are essential rules that help us simplify complex expressions involving powers. These laws are particularly useful in scientific notation, allowing us to deal with very large or small numbers easily. Let's understand a crucial rule used in our problem: * When raising a power to another power, multiply the exponents. Mathematically, it is expressed as \[ (a^b)^c = a^{bc} \]. * Working with scientific notation, such expressions like \((a \times 10^b)^n\) should be handled as \(a^n \times 10^{bn}\).In the exercise given, we applied this to both the numerator and the denominator. Such simplifications reduce the complexity of calculations. For example, in the numerator, \((3.542 \times 10^{-6})^9\) becomes \(3.542^9 \times 10^{-54}\). Simplifying expressions first before actual calculations with a calculator makes the process more efficient.

Significant Figures

Significant figures in a number tell us about the precision of a measurement or calculation. They are critical when we perform operations like multiplication or division, and the outcome should reflect the least precise measurement used. * Counting significant figures can be aided by some basic rules: * Non-zero digits are always significant. * Any zeros between significant digits are also significant. * Leading zeros in decimal numbers are not significant. * Trailing zeros in a decimal number are significant. In our exercise, "3.542" has four significant figures, and "5.05" has three. According to significant figure rules in division, the result should match the lowest number of significant figures given in the problem, which is here 3. This means you must round the final result to three significant figures to maintain mathematical integrity.

Calculator Usage

Calculators are indispensable tools, especially when dealing with complex calculations like those involving powers and scientific notation. Understanding how to efficiently use a calculator can save time and reduce errors. * Always input expressions in the correct order. * Utilize scientific notation functionality if your calculator supports it. This might involve using a button labeled "EXP" or "EE," which indicates the power of ten.* Break down complex expressions into simpler parts. For instance, calculate smaller components like \(3.542^9\) first and then tackle the division.In this exercise, we handled large powers which could prove cumbersome manually. A calculator can directly compute \(3.542^9\) and \(5.05^{12}\), providing precise numerical values that you can easily use in the division step. Finally, rounding the calculated value to the correct number of significant figures using the calculator ensures that your result aligns with scientific precision requirements.