Problem 101 - Chemistry Matter and Change

Question

Perform the following operations. (Chapter 2\()\) \begin{equation} \begin{array}{l}{\text { a. } 10^{7} \times 10^{3}} \\ {\text { b. }\left(1.4 \times 10^{-3}\right) \times\left(5.1 . \times 10^{-5}\right)} \\ {\text { c. }\left(2 \times 10^{-3}\right) \times\left(4 \times 10^{5}\right)}\end{array} \end{equation}

Step-by-Step Solution

Answer

a. \(10^{10}\), b. \(7.14 \times 10^{-8}\), c. \(8 \times 10^{2}\).

Step 1: Apply the Laws of Exponents

For part a, we are given the operation \(10^7 \times 10^3\). In multiplication of terms with the same base, we add the exponents: \[10^7 \times 10^3 = 10^{7 + 3} = 10^{10}\].

Step 2: Multiply the Coefficients and Add Exponents

For part b, we have \((1.4 \times 10^{-3}) \times (5.1 \times 10^{-5})\). First, multiply the coefficients: \[1.4 \times 5.1 = 7.14\]. Then, add the exponents of 10: \[10^{-3 + (-5)} = 10^{-8}\]. The result is \[7.14 \times 10^{-8}\].

Step 3: Multiply the Coefficients and Add Exponents for Positive Exponent

For part c, we have \((2 \times 10^{-3}) \times (4 \times 10^5)\). Multiply the coefficients: \[2 \times 4 = 8\]. Then, add the exponents of 10: \[10^{-3 + 5} = 10^{2}\]. The result is \[8 \times 10^{2}\].

Key Concepts

Laws of Exponents

When dealing with exponents, there are certain rules that help simplify calculations. These are known as the "laws of exponents". One basic law is that when multiplying numbers with the same base, like

• \(a^m \times a^n = a^{m+n}\)
  

This means you add the exponents together. Another important law is the power of a power rule:

• \((a^m)^n = a^{m \times n}\)
  

These rules not only simplify calculations but also make complex expressions manageable. For example, in the exercise above, multiplying \[10^7 \times 10^3\] simplifies to \[10^{7+3} = 10^{10}\].
These laws allow for easier manipulation and reduction of seemingly complicated exponential expressions. Understanding these laws aids in more efficient problem solving.

Multiplication of Powers

The multiplication of powers involves combining figures with exponents through a multiplication operation. It can be split into two parts, which are crucial to achieve a simplified result.First, you need to multiply the coefficients (the numerical parts). For instance, in \((1.4 \times 10^{-3}) \times (5.1 \times 10^{-5})\), the coefficients are 1.4 and 5.1. Multiplying them gives:\[1.4 \times 5.1 = 7.14\].
Next, apply the laws of exponents to the exponential parts. Specifically, add the exponents:\[10^{-3} \times 10^{-5} = 10^{-3+(-5)} = 10^{-8}\].
Thus, the final result is \[7.14 \times 10^{-8}\].
This step-by-step approach keeps the calculations organized and prevents errors when combining powers.

Scientific Notation

Scientific notation is a way to express extremely large or small numbers in a concise and accessible form. It is especially useful in science, engineering, and mathematics as it keeps numbers handy and readable by using a base of 10.A number in scientific notation is written in the form:

• \(a \times 10^n\),

where \(1 \leq a < 10\) and \(n\) is an integer. This notation makes calculations involving large values, such as \((2 \times 10^{-3}) \times (4 \times 10^{5})\),more manageable. Here, the coefficients, 2 and 4, are multiplied:\[2 \times 4 = 8\].
The exponents are added:\[10^{-3+5} = 10^{2}\],resulting in:\[8 \times 10^2\].
Scientific notation not only simplifies the computations but also helps in maintaining precision in calculations.