Final answer:
To order the numbers (1.1)^2, (1.4)^10, and (1.5)^16 from smallest to largest without doing precise calculations, we can use the understanding that for numbers greater than 1, the greater the base and the higher the exponent, the larger the result. Hence, (1.1)^2 is the smallest, followed by (1.4)^10, with (1.5)^16 being the largest.
Step-by-step explanation:
To order the numbers (1.1)^2, (1.4)^10, and (1.5)^16 from smallest to largest, we need to calculate the values of each. Since direct computation might be complex, we can use estimation or logarithms to compare the sizes of these numbers indirectly. However, without doing the actual calculations and given that raising a number greater than 1 to a higher power will result in a larger number, we can infer that as the base increases, so does the overall value if the exponents are of similar range.
Comparing the bases, we can see that 1.1 is the smallest and 1.5 is the largest. Taking the exponents into consideration, although 1.1 is raised to the power of 2, it is very unlikely that it will exceed or come close to the other two numbers which have much higher exponents. Therefore, we can confidently say that (1.1)^2 is the smallest. Between (1.4)^10 and (1.5)^16, even though the exponent for 1.4 is lesser, the difference in base suggests that (1.5)^16 will be significantly larger because the exponent is high enough to amplify the difference in base, indicating that (1.4)^10 is the next largest, with (1.5)^16 being the largest of the three.
Thus, the numbers in order from smallest to largest are:
- (1.1)^2
- (1.4)^10
- (1.5)^16