Which is larger? - - - - my challenge problem

[lookagain]

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Without using a calculator or computer, show all of the steps in your chosen method to determine which side has the larger value:


$\displaystyle \sqrt{7} \ + \ \sqrt[3]{13} \ \ \ \ versus \ \ \ \ 5$

 

[daon2]

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Without using a calculator or computer, show all of the steps in your chosen method to determine which side has the larger value:


$\displaystyle \sqrt{7} \ + \ \sqrt[3]{13} \ \ \ \ versus \ \ \ \ 5$

I'm sure there's a clever solution to this, but I went through the first way I thought of. All steps can be done by hand (but would take a long time).

Start with $\displaystyle [1.058301]^2/[0.4]^2 = 1.120001006601/0.16$, which is equal to $\displaystyle 7.000006+R_1$ for $\displaystyle R_1>0$.

So $\displaystyle \sqrt{7} < 1.058301/0.4 = 2.6457524$

Similarly, $\displaystyle 1.175667344^3/0.5^3 = 13.000000004+R_2$ where $\displaystyle R_2>0$.

So $\displaystyle \sqrt[3]{13} < 1.175667344/0.5 = 2.351334688$

$\displaystyle \sqrt{7} + \sqrt[3]{13} < 2.6457524+2.351334688 < 2.646+2.352 = 4.998 < 5$

 

[lookagain]

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.

Without using a calculator or computer, show all of the steps in your chosen method to determine which side has the larger value:


$\displaystyle \sqrt{7} \ + \ \sqrt[3]{13} \ \ \ \ versus \ \ \ \ 5$

Let's see if I have enough preliminary information:


1) $\displaystyle \ \$ For a and b belonging to the set of real numbers, and 0 < a < b,

$\displaystyle if \ \ a < b, \ \ then \ \ a^2 < b^2, \ \ and \ \ also \ \ a^3 < b^3.$


2) $\displaystyle \ \$ $\displaystyle (a - b)^3 \ = \ a^3 - 3a^2b + 3ab^2 - b^3$


(I am using "vs." for "versus.")


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$\displaystyle \sqrt{7} \ + \ \sqrt[3]{13} \ \ \ \ vs. \ \ \ \ 5$

$\displaystyle \sqrt[3]{13} \ \ \ \ vs. \ \ \ \ 5 - \sqrt{7}$

$\displaystyle (\sqrt[3]{13})^3 \ \ \ \ vs. \ \ \ \ (5 - \sqrt{7})^3$

$\displaystyle 13 \ \ \ \ vs. \ \ \ \ (5)^3 \ - \ 3(5)^2(\sqrt{7}) \ + \ 3(5)(\sqrt{7})^2 \ - \ (\sqrt{7})^3$

$\displaystyle 13 \ \ \ \ vs. \ \ \ \ 125 \ - \ 75\sqrt{7} \ + \ 105 \ - \ 7\sqrt{7}$

$\displaystyle 13 \ \ \ \ vs. \ \ \ \ 230 \ - \ 82\sqrt{7}$

$\displaystyle 82\sqrt{7} \ \ \ \ vs. \ \ \ \ 217$

$\displaystyle (82\sqrt{7})^2 \ \ \ \ vs. \ \ \ \ (217)^2$

$\displaystyle (82)^2(7) \ \ \ \ vs. \ \ \ \ (217)^2$

$\displaystyle 47,068 \ \ \ \ vs. \ \ \ \ 47,089$



Because the quantity on the right-hand side is greater than the quantity on the
left-hand side, then we conclude that:

$\displaystyle 5 \ > \ \sqrt{7} \ + \ \sqrt[3]{13}$