Question - Simplifying Exponential Expressions

To solve the expression \(10^{-8}\), we will simplify this using the properties of exponents. Let's break down the steps:

Step 1: Understanding Negative Exponents

The expression \(10^{-8}\) indicates that we have a base of 10 raised to a negative exponent. In general, any number \(a\) raised to a negative exponent \(n\) is expressed as:

This means that \(10^{-8}\) can be rewritten using this property.

Step 2: Applying the Negative Exponent Rule

Using the rule mentioned above, we can rewrite \(10^{-8}\) as follows:

This transformation is crucial as it moves the negative exponent to a positive exponent in the denominator.

Step 3: Understanding the Result

Now, we have simplified the expression to \( \frac{1}{10^8} \). This notation signifies that \(10^8\) (which equals 100,000,000) is in the denominator. Hence, we can express this in decimal form as follows:

Step 4: Final Result

The simplified result of \(10^{-8}\) is therefore:

In conclusion, the negative exponent transforms into a fraction with the base in the denominator, yielding a small decimal number. Understanding this property helps simplify expressions involving negative exponents effectively.