- Represent the given numbers in scientific notation: \(3.47866 \times 10^6\), \(3.487 \times 10^7\), and \(3.47866 \times 10^7\).
- Compare the exponents to identify that numbers with \(10^7\) are larger than the one with \(10^6\).
- Compare the coefficients \(3.487\) and \(3.47866\) to find the largest value.
- Identify that \(34,870,000\) (or \(3.487 \times 10^7\)) is the greatest.
- The correct choice is \(\boxed{3.487 \times 10^7}\).
Explanation
- Analyzing the Numbers
To compare the numbers \(3,478,660\), \(34,870,000\), and \(34,786,600\), we use scientific notation, which expresses a number as \(a \times 10^n\) where \(1 \le |a| < 10\) and \(n\) is an integer. Comparing numbers in this format is efficient: we first look at the exponent \(n\), and if the exponents are equal, we compare the coefficient \(a\).
- Converting to Scientific Notation
Now, we convert each number into scientific notation:
- For \(3,478,660\): We move the decimal point 6 places to the left, resulting in \(3.47866 \times 10^6\).
- For \(34,870,000\): We move the decimal point 7 places to the left, resulting in \(3.487 \times 10^7\).
- For \(34,786,600\): We move the decimal point 7 places to the left, resulting in \(3.47866 \times 10^7\).
- Comparing and Identifying the Greatest Number
Comparing the exponents:
- The first number has an exponent of \(6\), while the other two have an exponent of \(7\). Thus, \(3,478,660\) is the smallest.
- Between \(3.487 \times 10^7\) and \(3.47866 \times 10^7\), both have the same exponent (\(7\)). We compare the coefficients: \(3.487\) is greater than \(3.47866\).
- Therefore, \(34,870,000\) is the greatest number. In scientific notation, this is written as \(3.487 \times 10^7\).
Examples
Scientific notation is widely used by astronomers to represent the vast distances between stars and planets. For instance, the distance to the star Proxima Centauri is approximately \(4 \times 10^{16}\) meters. By using scientific notation, we can easily compare such massive numbers, perform complex calculations, and represent data in a concise way without dealing with long strings of zeros.