- Identify the first two significant figures for each number and use the third digit to round correctly.
- Convert each rounded value into its full numerical form, adjusting by the given powers of ten.
- Result: a) \(\boxed{3,200,000,000}\), b) \(\boxed{24,000,000}\), c) \(\boxed{16,000,000}\), d) \(\boxed{0.16}\), e) \(\boxed{0.0000022}\), f) \(\boxed{150,000,000}\).
Explanation
- Problem Analysis and Rounding Principles
We are given six values and asked to round each to two significant figures and then write them out in their full, expanded numerical form. The process of rounding to two significant figures involves identifying the first two non-zero digits of a number and examining the third digit to decide whether to round up or keep the current value. If the third digit is \(5\) or greater, we increment the second digit by one; otherwise, the first two digits remain as they are. Finally, we adjust the magnitude of the number based on its given order of magnitude (e.g., billion, million, or powers of ten).
- Rounding and Expansion (a, b, c)
A3.18 billion: The first two significant figures are \(3.1\). The third digit is \(8\). Since \(8 \geq 5\), we round up the second digit, giving \(3.2\). Thus, \(3.2\) billion is \(3,200,000,000\).
B23.9 million: The first two significant figures are \(23\). The third digit is \(9\). Since \(9 \geq 5\), we round up the second digit, giving \(24\). Thus, \(24\) million is \(24,000,000\).
C16.06 million: The first two significant figures are \(16\). The third digit is \(0\). Since \(0 < 5\), we keep the second digit as is, giving \(16\). Thus, \(16\) million is \(16,000,000\).
- Rounding and Expansion (d, e, f)
D\(1.5899 \times 10^{-1}\): The first two significant figures are \(1.5\). The third digit is \(8\). Since \(8 \geq 5\), we round up to \(1.6\). So, \(1.6 \times 10^{-1} = 0.16\).
E\(2.15 \times 10^{-6}\): The first two significant figures are \(2.1\). The third digit is \(5\). Since \(5 \geq 5\), we round up to \(2.2\). So, \(2.2 \times 10^{-6} = 0.0000022\).
f) \(1.496 \times 10^8\): The first two significant figures are \(1.4\). The third digit is \(9\). Since \(9 \geq 5\), we round up to \(1.5\). So, \(1.5 \times 10^8 = 150,000,000\).
- Final Answer summary
We have successfully rounded all six values to two significant figures and expressed them in their full numerical notation:
A\(\boxed{3,200,000,000}\)
B\(\boxed{24,000,000}\)
C\(\boxed{16,000,000}\)
D\(\boxed{0.16}\)
E\(\boxed{0.0000022}\)
f) \(\boxed{150,000,000}\)
Examples
Rounding large numbers is essential in fields like geography and science for clear communication. For example, if a researcher is reporting the population of different cities, they might use rounding to make the numbers easier to compare and present to the public. Similarly, when scientists describe the distance between planets, they often use rounded values in scientific notation to highlight the scale of the universe without getting lost in overly precise digits that may not be necessary for a general overview. Understanding how to round correctly ensures that data remains accurate enough for its purpose while being simple to understand and interpret.