Grades 9-10 Video Solutions 2010-Levels 9&10 Video Solutions 2010 problem10

Levels 9&10 Video Solutions 2010 problem10

Video Transcription

Question number 10. If a is equal to the square root of 2010, b is equal to 8 times the square  root of 30, and c is equal to the square root of 10 times 3 to the 5th power, then how are these numbers ordered with respect to one another? So let's just compare them as best  as we can. We have 2010 here, and the radical here, so to take that square root and estimate it, it's a good idea to know how to factor the contest year 2010. And from the previous  question, we might recall that this is equal to 10 times 201, which is 2 times 3 times 5 times 67. And so we have here 67, which is bigger than 64. So this is larger than  2 times 3 times 5 times 64, but that quantity is then equal to 30 times 64, like so. And we can even write 30 times 8 squared. Now, b is equal to 80 times the square root of  30, and we see that the square root of 2010, so versus the square root of 2010, which would be greater than the square root of 30 times 64, which is then equal to 8 times the square  root of 30. So we have here that a is bigger than b. And now we have to look at c. c is equal to the square root of 10 times 3 to the 5th power, like that, which is equal to  what we would like is to see a 30 in the radical. So 10 times 3 times 3 to the 4th power. And that is equal to 30 times 3 to the 4th power. And we can factor out a square here. So 3  squared times 30. And that is 9 times the square root of 30, which clearly is bigger than 8 times  the square root of 30. And that is exactly a. So we know that c is larger than a. And since a is  larger than b, then what we have here is our answer. c is the largest, a is in the middle,  and then b is smallest. And that is exactly what we see in d except ordered in reverse. b is the smallest number, a is in the middle, and c is the greatest.

Video Summary

The problem involves determining the order of three expressions involving square roots: \( a = \sqrt{2010} \), \( b = 8\sqrt{30} \), and \( c = \sqrt{10} \times 3^5 \). By comparing these, it's found that \( a \) is larger than \( b \), and \( c \) is larger than \( a \). Therefore, the order from smallest to largest is \( b < a < c \).