Difference between revisions of "2003 AMC 10A Problems/Problem 9"
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| − | <math>\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}=\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}=\sqrt[3]{x\sqrt{x}}= \sqrt{x} \Rightarrow A</math> | + | <math>\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}=\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}=\sqrt[3]{x\sqrt{x}}= \sqrt{x} \Rightarrow \boxed{\mathrm{(A)}\ \sqrt{x}}</math> |
== See Also == | == See Also == | ||
Revision as of 20:48, 3 January 2014
Problem
Simplify
![$\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}$](http://latex.artofproblemsolving.com/c/f/2/cf2756b7e9bb76b2adf76f1709cc765535b27602.png)
.
![$\mathrm{(A) \ } \sqrt{x}\qquad \mathrm{(B) \ } \sqrt[3]{x^{2}}\qquad \mathrm{(C) \ } \sqrt[27]{x^{2}}\qquad \mathrm{(D) \ } \sqrt[54]{x}\qquad \mathrm{(E) \ } \sqrt[81]{x^{80}}$](http://latex.artofproblemsolving.com/2/e/1/2e12822d4d418ef37b7ca03cff39d5b6e8123508.png)
Solution
![$\sqrt[3]{x\sqrt{x}}=\sqrt[3]{(\sqrt{x})^{2}\cdot\sqrt{x}}=\sqrt[3]{(\sqrt{x})^{3}}=\sqrt{x}$](http://latex.artofproblemsolving.com/5/b/5/5b532d915bdc855d96cfa266d7d3bb9a168fd803.png)
.
Therefore:
![$\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}=\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}=\sqrt[3]{x\sqrt{x}}= \sqrt{x} \Rightarrow \boxed{\mathrm{(A)}\ \sqrt{x}}$](http://latex.artofproblemsolving.com/7/a/6/7a6a1831eb969273f9c00ce0ca3ba2e52d2b5c72.png)
See Also
| 2003 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 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 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
