When the numbers $\sqrt{2}, \sqrt[3]{3},$ and $\sqrt[5]{5}$ are listed in order from least to greatest, which number is in the middle?

Suppose $a$ and $b$ are positive integers and $\dfrac{a}{4}+\dfrac{b}{3}=\dfrac{13}{12}$. What is the value of $a^2+b^2$?

What is the largest perfect square less than 12345654320?

How many positive integers less than 2000 are of the form $x^n$ for some positive integer $x$ and $n\ge 2?$

I have until next Saturday, so take your time!

Imnotamaster Sep 15, 2019

#1**+1 **

When the numbers \($\sqrt{2}, \sqrt[3]{3},$ and $\sqrt[5]{5}$ \) are listed in order from least to greatest, which number is in the middle?

Compare

√2 ^{3}√3 , ^{5}√5 raise each to the 30th power

(2^(1/2))^30 (3^(1/3))^30 (5^(1/5))^30

2^15 3^10 5^6

32768 59049 15625

So

√2 is the middle number

CPhill Sep 15, 2019

#2**+1 **

Suppose a and b are positive integers and \(\dfrac{a}{4}+\dfrac{b}{3}=\dfrac{13}{12}\). What is the value of a^2+b^2?

a/4 + b/3 = 13/12

[ 3a + 4b] 13

_______ = ___

12 12

This implies that 3a + 4b = 13

If we want positive integer values.....then a = 3 and b = 1

So

a^2 + b^2 =

3^2 + 1^2 =

9 + 1 =

10

CPhill Sep 15, 2019

#3**+1 **

Thank you so much, c-phil!

Ironically, just as you answered one of the questions, I figured out one of them too.

Imnotamaster Sep 15, 2019