Grades 11-12 Video Solutions 2013-Grades 11-12 Video Solutions 2013 problem1

Video Transcription

Question number one, which of the following numbers is the largest?  Let's begin by considering two of the smallest numbers.  We have 20 times 13 equal to 260, which is smaller than 2013.  Now, to obtain estimates for the remaining numbers, we rewrite them as powers of 2 whenever possible. So, recalling that 2 to the power 10 is 1024,  we have an upper bound for 2013, 2048,  which is 2 to the 11th power. And that's less than 2 to the power 13.  Now, by increasing the base from 2 to 20, we obtain a larger number still.  And so, it remains for us to estimate the number in D. A lower bound for 201 would be 128.  And that's 2 to the power 7.  So, in base 2, we have a lower bound of 2 to the power 21.  Now, an upper bound for 201 is 256,  which is equal to 2 to the 8th power.  And so, we have an upper bound of 2 to the power 24.  Where does that fit in in comparison to 20 to the power 13?  We can obtain a lower bound  by rewriting the exponent as 2 plus 11.  So, 20 squared is 400 to the 11th power. And then, lower bound is, again, 256. So, we write 2 to the 8th power, 2 to the 11th, and obtain a lower bound of 2 to the power 88.  Clearly larger than 2 to the power 24.  So, the number in C is the largest of these numbers.

Video Summary

The largest number among the options is determined by comparing numbers converted or estimated as powers of 2. Starting with 20 times 13 equals 260, which is smaller than 2013, and estimating other numbers as powers of 2 shows that 2013 is less than 2 to the 11th power (2048). When comparing against a higher base, 20 to the power 13 gives a lower bound around 2 to the 88th power, suggesting it is significantly larger than the other options. Thus, option C (20 to the power 13) is identified as the largest number.

Keywords

powers of 2

largest number

comparison

exponentiation

estimation