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Which is larger? - - - - my challenge problem
- Thread starter lookagain
- Start date
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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 \(\displaystyle \sqrt[3]{13} < 1.175667344/0.5 = 2.351334688\)
\(\displaystyle \sqrt{7} + \sqrt[3]{13} < 2.6457524+2.351334688 < 2.646+2.352 = 4.998 < 5\)
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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:
\(\displaystyle 5 \ > \ \sqrt{7} \ + \ \sqrt[3]{13}\)
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