- Observe the table pattern: positive powers of 10 represent whole numbers, while negative powers represent fractions with 1 in the numerator and powers of 10 in the denominator.
- Identify that \(1,000,000\) is equal to \(10^6\).
- Apply the rule \(\frac{1}{10^k} = 10^{-k}\) to rewrite \(\frac{1}{1,000,000}\) as \(10^{-6}\).
- Match the result to option b: \(\boxed{b}\).
Explanation
- Analysis of the Pattern
We are given a table illustrating powers of 10 and their corresponding values:
\[ \begin{array}{|l|l|} \ \text{Powers of 10} & \text{Value} \\ 10^1 & 10 \\ 10^0 & 1 \\ 10^{-1} & \frac{1}{10} \\ 10^{-2} & \frac{1}{100} \\ 10^{-3} & \frac{1}{1000} \\ 10^{-4} & \frac{1}{10,000} \\ \end{array} \]
The goal is to determine the exponent \(n\) such that \(10^n = \frac{1}{1,000,000}\). By observing the pattern, we see that for any positive integer \(k\), the value of \(10^{-k}\) is \(\frac{1}{10^k}\). Our task is to write the fraction \(\frac{1}{1,000,000}\) in this form.
- Determining the Exponent
We need to identify the power of 10 that equals \(1,000,000\). By counting the number of zeros in \(1,000,000\), we find there are 6 zeros. Thus, \(1,000,000 = 10^6\). Using the pattern identified earlier where \(\frac{1}{10^k} = 10^{-k}\), we can substitute \(k = 6\) into this expression. This gives us:
\[ \frac{1}{1,000,000} = \frac{1}{10^6} = 10^{-6} \]
Comparing this result to the given options:
A\(10^6\)
B\(10^{-6}\)
C\(10^{-5}\)
The calculated value matches option b.
- Conclusion
Based on the analysis, the power of 10 that represents \(\frac{1}{1,000,000}\) is \(10^{-6}\). The steps taken show that the number of zeros in the denominator of the fraction corresponds to the absolute value of the negative exponent, confirmable through the power of 10 property \(\frac{1}{10^k} = 10^{-k}\). Thus, the correct option is \(\boxed{b}\).
Examples
In science and engineering, we often deal with extremely small measurements, such as the size of a bacteria or the weight of a microscopic particle. For instance, if a particle weighs \(\frac{1}{1,000,000}\) of a gram, it is much easier to express this weight using negative scientific notation as \(10^{-6}\) grams. This notation helps scientists avoid writing many zeros and simplifies calculations when working with very small quantities.