Final answer:
Truncating numbers after the eighth decimal place before squaring them will ensure the results are within 10^-7 of the true value because the largest potential error, (10^-8)^2, results in an error less than 10^-7.
Step-by-step explanation:
To determine if truncating after the eighth place of decimal expansions ensures that the outputs are within 10⁻⁷ of the true value, let's consider the worst-case scenario. This would occur with the largest number in our range, which is just below 10. If we truncate this number at the eighth decimal place and then square it, the effect of the last included digit (in the ninth decimal place) on the squared value is at most (10⁻⁸)², which equals 10⁻‖. This is because when squaring a number, the maximum error is squared along with the number. Since 10⁻‖ is less than 10⁻⁷, we can conclude that truncating the number prior to squaring it at the eighth decimal place will indeed ensure the squared result is within 10⁻⁷ of the true value.
Additionally, the concept of squaring numbers and understanding decimal places is closely related to the properties of exponents, like those applied in scientific notation and exponentiation. When we square a number in scientific notation, we multiply the exponent by two. For example, (4.506 × 10⁴)² becomes (4.506²) × (10⁴)² = 4.506² × 10⁸. This understanding helps us realize the magnitudes of the errors involved in this process.