Problem 71 - Elementary Algebra

Question

\(\frac{1}{4 \times 10^{8}}\)

Step-by-Step Solution

Answer

0.25 \times 10^{-8}

Step 1 - Understand the Problem

The problem requires simplifying the expression \(\frac{1}{4 \times 10^{8}}\). This involves simplifying the fraction and exploring the multiplication of the number by a power of ten.

Step 2 - Simplify the Denominator

The denominator in the expression is \(4 \times 10^{8}\). Recognize that this can be broken into two separate components: 4 and \(10^{8}\).

Step 3 - Invert the Denominator

To simplify the fraction, divide 1 by each component separately. This gives you \(\frac{1}{4} \times \frac{1}{10^{8}}\).

Step 4 - Simplify Each Fraction

Firstly, \( \frac{1}{4} = 0.25 \). Secondly, \( \frac{1}{10^{8}} = 10^{-8} \). Therefore, we have: \(\frac{1}{4 \times 10^{8}} = 0.25 \times 10^{-8} \).

Step 5 - Combine Components

Finally, combine the result from the previous step: \(0.25 \times 10^{-8} \).

Key Concepts

powers of ten

When dealing with large or small numbers, it is common to use powers of ten. A power of ten is an expression like \(\text{10}^n\), where n is an exponent. If n is positive, it represents a big number (e.g., \(\text{10}^3 = 1000\)). If n is negative, it represents a fraction (e.g., \(\text{10}^{-3} = 0.001\)).

Powers of ten allow us to easily write and calculate with very large or small numbers. For example, \(10^8\) is 100,000,000. It’s easy to see how multiplying or dividing by powers of ten can affect a number. To simplify calculations, we can manage these exponents separately from other parts of an expression.

fraction division

Let's break down fraction division using our example \(\frac{1}{4 \times 10^8}\). First, understand that we are dividing 1 by the product of 4 and \(10^8\). To make this easier, we can split the denominator: \(4 \times 10^8\) becomes two separate parts: 4 and \(10^8\).

This helps us divide 1 by each part individually. \( \frac{1}{4} \) is simplified to 0.25. And \( \frac{1}{10^8} \) can be written as \(10^{-8}\). By combining these simplified parts, we get \(0.25 \times 10^{-8}\). Understanding this step-by-step breaking down makes fraction division much simpler.

scientific notation

Scientific notation is a method to express very large or very small numbers. It’s written as the product of a number between 1 and 10 and a power of ten. For example, 500,000 is written as \(5 \times 10^5\) in scientific notation.

This notation is particularly helpful for readability and calculations. In our problem, we used scientific notation when we converted \(\frac{1}{10^8}\) to \(\text{10}^{-8}\). This helps simplify the expression. When we combined the results, we got \(0.25 \times 10^{-8} \), which is easy to understand and work with. Using scientific notation simplifies complex numbers and calculations.