Difference between revisions of "2011 AMC 10A Problems/Problem 16"
(added a better solution) |
(added a better solution) |
||
Line 18: | Line 18: | ||
(Basically, this method turns the question into a 4th root and then simplifies it. By the way, this method is much easier.) | (Basically, this method turns the question into a 4th root and then simplifies it. By the way, this method is much easier.) | ||
+ | Request from the author: Can someone help fix the coding? | ||
+ | Thx ------ SuperWill | ||
− | + | == Solution 3 (FASTER!) == | |
− | |||
− | == Solution 3 (FASTER!== | ||
We can change the insides of the square root into a perfect square and then simplify. | We can change the insides of the square root into a perfect square and then simplify. | ||
− | < | + | <math>&\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}\\ = \ &\sqrt{6-6\sqrt{2}+3}+\sqrt{6+6\sqrt{2}+3\\ = \ &\sqrt{\left{\sqrt{6}-\sqrt{3}}^2}+\sqrt{\left{\sqrt{6}+\sqrt{3}}^2}\\ = \ &\sqrt{6}-\sqrt{3}+\sqrt{6}+\sqrt{3}\\ = \ &\boxed{2\sqrt{6} \ \mathbf{(B)}}</math> |
− | &\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}\\ = \ &\sqrt{6-6\sqrt{2}+3}+\sqrt{6+6\sqrt{2}+3\\ = \ &\sqrt{\left{\sqrt{6}-\sqrt{3}}^2}+\sqrt{\left{\sqrt{6}+\sqrt{3}}^2}\\ = \ &\sqrt{6}-\sqrt{3}+\sqrt{6}+\sqrt{3}\\ = \ &\boxed{2\sqrt{6} \ \mathbf{(B)} | ||
− | |||
-----will3145 | -----will3145 | ||
== See Also == | == See Also == |
Revision as of 20:40, 8 July 2018
Problem 16
Which of the following is equal to ?
Solution 1
We find the answer by squaring, then square rooting the expression.
Solution 2
\begin{align*} &\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}\\ = \ &\sqrt{\left(\sqrt{81-38}+\sqrt{81+38}\right)}\\ = \ &\sqrt{\left(\sqrt{162}\right}\\ = \ &\sqrt{\left(\sqrt{(3^4)*2\right} = \ &\boxed{2\sqrt{6} \ \mathbf{(B)}}. \end{align*} (Error compiling LaTeX. Unknown error_msg)
(Basically, this method turns the question into a 4th root and then simplifies it. By the way, this method is much easier.) Request from the author: Can someone help fix the coding? Thx ------ SuperWill
Solution 3 (FASTER!)
We can change the insides of the square root into a perfect square and then simplify.
$&\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}\\ = \ &\sqrt{6-6\sqrt{2}+3}+\sqrt{6+6\sqrt{2}+3\\ = \ &\sqrt{\left{\sqrt{6}-\sqrt{3}}^2}+\sqrt{\left{\sqrt{6}+\sqrt{3}}^2}\\ = \ &\sqrt{6}-\sqrt{3}+\sqrt{6}+\sqrt{3}\\ = \ &\boxed{2\sqrt{6} \ \mathbf{(B)}}$ (Error compiling LaTeX. Unknown error_msg)
will3145
See Also
2011 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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.