Difference between revisions of "2011 AMC 10A Problems/Problem 16"
m (→Solution 2) |
(added a better solution) |
||
| Line 13: | Line 13: | ||
== Solution 2 == | == Solution 2 == | ||
| − | < | + | <cmath>\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)}}. | &\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*}</ | + | \end{align*}</cmath> |
| − | (Basically this method turns the question into a 4th root | + | (Basically, this method turns the question into a 4th root and then simplifies it. By the way, this method is much easier.) |
| − | + | ------ SuperWill | |
| − | |||
| + | == Solution 3 (FASTER!== | ||
| + | We can change the insides of the square root into a perfect square and then simplify. | ||
| + | |||
| + | <cmath>\begin{align*} | ||
| + | &\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)}}. | ||
| + | \end{align*}</cmath> | ||
| + | -----will3145 | ||
== See Also == | == See Also == | ||
Revision as of 19:38, 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.)
SuperWill
Solution 3 (FASTER!
We can change the insides of the square root into a perfect square and then simplify.
\begin{align*}
&\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)}}.
\end{align*} (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. 
