Difference between revisions of "1991 AHSME Problems/Problem 6"
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== Problem == | == Problem == | ||
| − | |||
If <math>x\geq 0</math>, then <math>\sqrt{x\sqrt{x\sqrt{x}}}=</math> | If <math>x\geq 0</math>, then <math>\sqrt{x\sqrt{x\sqrt{x}}}=</math> | ||
| − | (A) | + | <math>\textbf{(A) } x\sqrt{x}\qquad |
| + | \textbf{(B) } x\sqrt[4]{x}\qquad | ||
| + | \textbf{(C) } \sqrt[8]{x}\qquad | ||
| + | \textbf{(D) } \sqrt[8]{x^3}\qquad | ||
| + | \textbf{(E) } \sqrt[8]{x^7}</math> | ||
== Solution == | == Solution == | ||
| − | <math>\ | + | Recall that square roots are one-half powers, namely <math>\sqrt y=y^{\frac12}</math> for all <math>y\geq0.</math> |
| + | |||
| + | We have | ||
| + | <cmath>\begin{align*} | ||
| + | \sqrt{x\sqrt{x\sqrt{x}}} &= \sqrt{x\sqrt{x\cdot x^{\frac12}}} \\ | ||
| + | &= \sqrt{x\sqrt{x^{\frac32}}} \\ | ||
| + | &= \sqrt{x\cdot\left(x^{\frac32}\right)^{\frac12}} \\ | ||
| + | &= \sqrt{x^{\frac74}} \\ | ||
| + | &= \left(x^{\frac74}\right)^{\frac12} \\ | ||
| + | &= x^{\frac78} \\ | ||
| + | &= \boxed{\textbf{(E) } \sqrt[8]{x^7}}. | ||
| + | \end{align*}</cmath> | ||
| + | ~Hapaxoromenon (Solution) | ||
| + | |||
| + | ~MRENTHUSIASM (Reformatting) | ||
== See also == | == See also == | ||
Latest revision as of 18:13, 5 September 2021
Problem
If 
, then 
![$\textbf{(A) } x\sqrt{x}\qquad \textbf{(B) } x\sqrt[4]{x}\qquad \textbf{(C) } \sqrt[8]{x}\qquad \textbf{(D) } \sqrt[8]{x^3}\qquad \textbf{(E) } \sqrt[8]{x^7}$](http://latex.artofproblemsolving.com/b/c/a/bca02acf1c63a8ddb1663dd2831f63615ad8c4de.png)
Solution
Recall that square roots are one-half powers, namely 
for all 
We have
![\begin{align*} \sqrt{x\sqrt{x\sqrt{x}}} &= \sqrt{x\sqrt{x\cdot x^{\frac12}}} \\ &= \sqrt{x\sqrt{x^{\frac32}}} \\ &= \sqrt{x\cdot\left(x^{\frac32}\right)^{\frac12}} \\ &= \sqrt{x^{\frac74}} \\ &= \left(x^{\frac74}\right)^{\frac12} \\ &= x^{\frac78} \\ &= \boxed{\textbf{(E) } \sqrt[8]{x^7}}. \end{align*}](http://latex.artofproblemsolving.com/1/2/8/1280bb820f79bcf16e11c8fb87b4fa71037b3dd8.png)
~Hapaxoromenon (Solution)
~MRENTHUSIASM (Reformatting)
See also
| 1991 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 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. 
