- Use scientific notation properties: a larger exponent implies a larger number, and for equal exponents, a larger coefficient implies a larger number.
- Compare each option against \(1.429 \times 10^{-4}\):
- \(4.129 \times 10^{-4} > 1.429 \times 10^{-4}\) (higher)
- \(1.429 \times 10^{-6} < 1.429 \times 10^{-4}\) (lower)
- \(1.429 \times 10^{-5} < 1.429 \times 10^{-4}\) (lower)
- \(1.429 \times 10^{-3} > 1.429 \times 10^{-4}\) (higher)
- Conclude that both \(\boxed{4.129 \times 10^{-4}}\) and \(\boxed{1.429 \times 10^{-3}}\) are higher than the given value.
Explanation
- Analysis of Scientific Notation Comparison
We are comparing the given density of oxygen, \(1.429 \times 10^{-4}\), with several other numbers in scientific notation. Scientific notation represents a number as \(a \times 10^n\), where \(1 \leq |a| < 10\) and \(n\) is an integer. To compare two positive numbers in this form, we first look at the exponent \(n\). A larger exponent \(n\) signifies a larger number. If the exponents are equal, we compare the leading coefficients \(a\). If \(a_1 > a_2\), then \(a_1 \times 10^n > a_2 \times 10^n\).
- Evaluation of Options
We evaluate each option:
- \(4.129 \times 10^{-4}\): The exponent is \(-4\), which is the same as the reference. Comparing the coefficients, \(4.129 > 1.429\), thus \(4.129 \times 10^{-4} > 1.429 \times 10^{-4}\).
- \(1.429 \times 10^{-6}\): The exponent \(-6 < -4\), therefore \(1.429 \times 10^{-6} < 1.429 \times 10^{-4}\).
- \(1.429 \times 10^{-5}\): The exponent \(-5 < -4\), therefore \(1.429 \times 10^{-5} < 1.429 \times 10^{-4}\).
- \(1.429 \times 10^{-3}\): The exponent \(-3 > -4\), therefore \(1.429 \times 10^{-3} > 1.429 \times 10^{-4}\).
Both the first and fourth options are mathematically higher. Assuming the question expects identification of all values satisfying the criteria, both \(4.129 \times 10^{-4}\) and \(1.429 \times 10^{-3}\) are higher. In standard testing contexts, both options represent correct choices.
Examples
Scientific notation is crucial in fields like chemistry and physics for expressing very small or very large quantities, such as the mass of an atom or the distance between planets. Being able to compare these values quickly allows scientists to determine which substances are denser, which distances are greater, or how concentration levels vary, facilitating efficient data analysis and research.