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# If you could help, that would be great!

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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!

Sep 15, 2019

#1
+106533
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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

Sep 15, 2019
#2
+106533
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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

Sep 15, 2019
#3
+124
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Thank you so much, c-phil!

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

Sep 15, 2019
#4
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What is the largest perfect square less than 12345654320?

Just take the square root of   12345654320  ≈  111110.9

So....the largest perfect square  =  (111110)^2 =   12345432100

Sep 15, 2019
#5
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For number four, I recommend proceeding with casework with n=2, n=3, n=4, etc.

Sep 15, 2019