Problem 35 - Algebra for College Students

Question

Use scientific notation and the properties of exponents to help you perform the following operations. \((0.00007)(11,000)\)

Step-by-Step Solution

Answer

\(7.7 \times 10^{-1}\)

Step 1: Express in Scientific Notation

Convert each number to scientific notation. - 0.00007 can be expressed as \(7 \times 10^{-5}\).- 11,000 can be expressed as \(1.1 \times 10^4\).

Step 2: Multiply the Coefficients

Multiply the coefficients (the numbers in front):\(7 \times 1.1 = 7.7\).

Step 3: Add the Exponents

Use the properties of exponents to add the exponents:\(10^{-5} \times 10^{4} = 10^{-5+4} = 10^{-1}\).

Step 4: Combine Results

Combine the results from Steps 2 and 3 to form the final result: \(7.7 \times 10^{-1}\).

Key Concepts

Properties of Exponents

When dealing with exponents, several rules can make calculations simpler and help us avoid errors. One essential property is that when you multiply two powers with the same base, you add their exponents.
For example, if you have two numbers such as \(10^a \times 10^b\), the result can be simplified to \(10^{a+b}\). This property allows us to handle even complex equations with ease, especially in scientific notation.
Another important rule is any number raised to the power of zero equals one. So, \(10^0 = 1\). Keep these properties in mind; they apply broadly and can simplify various mathematical operations.

Multiplication in Scientific Notation

Scientific notation is useful when dealing with very large or small numbers. You multiply numbers in scientific notation by separately multiplying their coefficients and adding their exponents.
Let's work through an example: \((7 \times 10^{-5})(1.1 \times 10^4)\).

• First, multiply the coefficients: \(7 \times 1.1 = 7.7\).
• Next, add the exponents: \(-5 + 4 = -1\).

Putting it together, you get \(7.7 \times 10^{-1}\). This process keeps calculations tidy and is especially useful in scientific and engineering contexts.

Converting to Scientific Notation

Converting numbers into scientific notation is about making them easier to read and manipulate. A number is typically written in the form of \(a \times 10^n\), where \(1 \leq a < 10\) and \(n\) is an integer.
For instance, to convert 0.00007 to scientific notation, we move the decimal point 5 places to the right, resulting in \(7 \times 10^{-5}\). The negative exponent indicates the decimal was shifted to the right.
Similarly, for a number like 11,000, shift the decimal point 4 places to the left to get \(1.1 \times 10^4\). It's a powerful tool that simplifies handling both very small and large numbers.

Exponents

Exponents are a shorthand way of expressing repeated multiplication. They consist of a base and a power, represented as \(b^n\). Here, the base \(b\) is multiplied by itself \(n\) times.
For example, \(2^3 = 2 \times 2 \times 2 = 8\), showing how exponents facilitate the expression of large numbers in compact form.

• Exponents speed up multiplication into a manageable process.
• For negative exponents like \(10^{-1}\), the interpretation is \(\frac{1}{10^1}\).

This allows us to transcribe and work with both vast and minuscule quantities efficiently, which is crucial in fields like science and mathematics.