Difference between revisions of "1998 AHSME Problems/Problem 7"
(Created page with "== Problem 7 == If <math>N > 1</math>, then <math>\sqrt[3]{N\sqrt[3]{N\sqrt[3]{N}}} =</math> <math> \mathrm{(A) \ } N^{\frac 1{27}} \qquad \mathrm{(B) \ } N^{\frac 1{9}} \qquad ...") |
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| − | == Problem | + | == Problem == |
If <math>N > 1</math>, then <math>\sqrt[3]{N\sqrt[3]{N\sqrt[3]{N}}} =</math> | If <math>N > 1</math>, then <math>\sqrt[3]{N\sqrt[3]{N\sqrt[3]{N}}} =</math> | ||
<math> \mathrm{(A) \ } N^{\frac 1{27}} \qquad \mathrm{(B) \ } N^{\frac 1{9}} \qquad \mathrm{(C) \ } N^{\frac 1{3}} \qquad \mathrm{(D) \ } N^{\frac {13}{27}} \qquad \mathrm{(E) \ } N</math> | <math> \mathrm{(A) \ } N^{\frac 1{27}} \qquad \mathrm{(B) \ } N^{\frac 1{9}} \qquad \mathrm{(C) \ } N^{\frac 1{3}} \qquad \mathrm{(D) \ } N^{\frac {13}{27}} \qquad \mathrm{(E) \ } N</math> | ||
| − | [[ | + | ==Solution== |
| + | The key identities are <math>\sqrt[3]{x^n} = x^{\frac{n}{3}}</math> and <math>x \cdot x^{\frac{a}{b}} = x^{1 + \frac{a}{b}}</math> | ||
| + | |||
| + | <math>\sqrt[3]{N\sqrt[3]{N\sqrt[3]{N}}}</math> | ||
| + | |||
| + | <math>\sqrt[3]{N\sqrt[3]{N\cdot N^{\frac{1}{3}}}}</math> | ||
| + | |||
| + | <math>\sqrt[3]{N\sqrt[3]{N^{\frac{4}{3}}}}</math> | ||
| + | |||
| + | <math>\sqrt[3]{N\cdot{N^{\frac{4}{9}}}}</math> | ||
| + | |||
| + | <math>\sqrt[3]{{N^{\frac{13}{9}}}}</math> | ||
| + | |||
| + | <math>N^{\frac{13}{27}}</math>, thus the answer is <math>\boxed{D}</math> | ||
| + | |||
| + | ==See Also== | ||
| + | {{AHSME box|year=1998|num-b=6|num-a=8}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 13:28, 5 July 2013
Problem
If 
, then ![$\sqrt[3]{N\sqrt[3]{N\sqrt[3]{N}}} =$](http://latex.artofproblemsolving.com/e/4/f/e4ffb8d1d2e875fc449d02088aee08a73f9b2079.png)
Solution
The key identities are ![$\sqrt[3]{x^n} = x^{\frac{n}{3}}$](http://latex.artofproblemsolving.com/d/e/e/deee5ab646d686f6ab1daf24c8f60861fb692094.png)
and 
![$\sqrt[3]{N\sqrt[3]{N\sqrt[3]{N}}}$](http://latex.artofproblemsolving.com/3/9/2/3928705993da8c6bc5863c4d4400db2099dd2d55.png)
![$\sqrt[3]{N\sqrt[3]{N\cdot N^{\frac{1}{3}}}}$](http://latex.artofproblemsolving.com/3/c/7/3c75c9eabc99a5fe7137418757e69ccb4635eae0.png)
![$\sqrt[3]{N\sqrt[3]{N^{\frac{4}{3}}}}$](http://latex.artofproblemsolving.com/a/c/9/ac92f5062a2cb1c13d0dc0c8d32f360ca2d7f00b.png)
![$\sqrt[3]{N\cdot{N^{\frac{4}{9}}}}$](http://latex.artofproblemsolving.com/3/5/3/353756add9e01230053920611b5834b6cb2eac59.png)
![$\sqrt[3]{{N^{\frac{13}{9}}}}$](http://latex.artofproblemsolving.com/f/9/a/f9a757658a98539dc0500a311d195ec2ac76b68d.png)
, thus the answer is 
See Also
| 1998 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
