2021 AMC 10B Problems/Problem 2: Difference between revisions
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==Problem== | ==Problem== | ||
What is the value of < | What is the value of <math>\sqrt{\left(3-2\sqrt{3}\right)^2}+\sqrt{\left(3+2\sqrt{3}\right)^2}</math>? | ||
<math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~4\sqrt{3}-6 \qquad\textbf{(C)} ~6 \qquad\textbf{(D)} ~4\sqrt{3} \qquad\textbf{(E)} ~4\sqrt{3}+6</math> | <math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~4\sqrt{3}-6 \qquad\textbf{(C)} ~6 \qquad\textbf{(D)} ~4\sqrt{3} \qquad\textbf{(E)} ~4\sqrt{3}+6</math> | ||
==Solution== | ==Solution 1== | ||
Note that the square root of | Note that the square root of any number squared is always the absolute value of the squared number because the square root function will only return a nonnegative number. By squaring both <math>3</math> and <math>2\sqrt{3}</math>, we see that <math>2\sqrt{3}>3</math>, thus <math>3-2\sqrt{3}</math> is negative, so we must take the absolute value of <math>3-2\sqrt{3}</math>, which is just <math>2\sqrt{3}-3</math>. Knowing this, the first term in the expression equals <math>2\sqrt{3}-3</math> and the second term is <math>3+2\sqrt3</math>, and summing the two gives <math>\boxed{\textbf{(D)} ~4\sqrt{3}}</math>. | ||
~bjc, abhinavg0627 and JackBocresion | |||
<math>\ | |||
==Solution 2== | |||
Let <math>x = \sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}</math>, then <math>x^2 = (3-2\sqrt{3})^2+2\sqrt{(-3)^2}+(3+2\sqrt3)^2</math>. The <math>2\sqrt{(-3)^2}</math> term is there due to difference of squares when you simplify <math>2ab</math> from <math>(a + b)^2</math>. Simplifying the expression gives us <math>x^2 = 48</math>, so <math>x=\boxed{\textbf{(D)} ~4\sqrt{3}}</math> ~ shrungpatel | |||
==Solution 3 (Memorizing Roots)== | |||
Memorizing your square roots from 1 - 10 are really important for cheesing some AMC problems, so try to memorize them. | |||
\( \sqrt{3} \) is about 1.7. | |||
We then substitute \( \sqrt{3} \) for 1.7 to solve this. | |||
We get | |||
<cmath> | |||
\sqrt{(3-2 \cdot 1.7)^2} + \sqrt{(3+2 \cdot 1.7)^2} | |||
</cmath> | |||
<cmath> | |||
= \sqrt{(-0.4)^2} + \sqrt{(6.4)^2} | |||
</cmath> | |||
<cmath> | |||
= \sqrt{0.16} + \sqrt{6.4^2} | |||
</cmath> | |||
<cmath> | |||
= 0.4 + 6.4 | |||
</cmath> | |||
<cmath> | |||
= 6.8 | |||
</cmath> | |||
Looking at the answer choices, we see that <math>\boxed{\textbf{(D)} ~4\sqrt{3}}</math> gives <math>6.8</math> for when \( \sqrt{3} = 1.7 \). | |||
~Pinotation | |||
==Video Solution== | |||
https://youtu.be/HHVdPTLQsLc | |||
~Math Python | |||
==Video Solution (EASY TO UNDERSTAND📈)== | |||
https://www.youtube.com/watch?v=A1Li_jciTZY | |||
~CalculaCore | |||
== Video Solution by OmegaLearn == | |||
https://youtu.be/Df3AIGD78xM | |||
~pi_is_3.14 | |||
==Video Solution== | |||
https://youtu.be/v71C6cFbErQ | |||
~savannahsolver | |||
==Video Solution by TheBeautyofMath== | |||
https://youtu.be/gLahuINjRzU?t=154 | |||
~IceMatrix | |||
==Video Solution by Interstigation== | |||
https://youtu.be/DvpN56Ob6Zw?t=1 | |||
~Interstigation | |||
==Video Solution by Mathematical Dexterity (50 Seconds)== | |||
https://www.youtube.com/watch?v=ScZ5VK7QTpY | |||
==Video Solution== | |||
https://youtu.be/3GHD62FK0xY | |||
~Education, the Study of Everything | |||
==See Also== | |||
{{AMC10 box|year=2021|ab=B|num-b=1|num-a=3}} | |||
{{MAA Notice}} | |||
Latest revision as of 13:55, 5 September 2025
Problem
What is the value of 
?
Solution 1
Note that the square root of any number squared is always the absolute value of the squared number because the square root function will only return a nonnegative number. By squaring both 
and 
, we see that 
, thus 
is negative, so we must take the absolute value of , which is just 
. Knowing this, the first term in the expression equals and the second term is 
, and summing the two gives 
.
~bjc, abhinavg0627 and JackBocresion
Solution 2
Let 
, then 
. The 
term is there due to difference of squares when you simplify 
from 
. Simplifying the expression gives us 
, so 
~ shrungpatel
Solution 3 (Memorizing Roots)
Memorizing your square roots from 1 - 10 are really important for cheesing some AMC problems, so try to memorize them.
\( \sqrt{3} \) is about 1.7.
We then substitute \( \sqrt{3} \) for 1.7 to solve this.
We get
![\[\sqrt{(3-2 \cdot 1.7)^2} + \sqrt{(3+2 \cdot 1.7)^2}\]](http://latex.artofproblemsolving.com/3/d/5/3d53fdecc00161231f7130f8c71fe94497648df5.png)
![\[= \sqrt{(-0.4)^2} + \sqrt{(6.4)^2}\]](http://latex.artofproblemsolving.com/0/e/8/0e8eebbbc9a8852d1b0a7eecb9f52fa86b05b8f6.png)
![\[= \sqrt{0.16} + \sqrt{6.4^2}\]](http://latex.artofproblemsolving.com/2/4/1/2413544acdf0909f6d93c51bb3e710eeaf65e911.png)
![\[= 0.4 + 6.4\]](http://latex.artofproblemsolving.com/a/c/c/acce90e2c132324f21babe9fd0f68cad20250ee8.png)
![\[= 6.8\]](http://latex.artofproblemsolving.com/2/f/2/2f22619805430e3bc6bc130a028a33fa8b16658f.png)
Looking at the answer choices, we see that gives 
for when \( \sqrt{3} = 1.7 \).
~Pinotation
Video Solution
https://youtu.be/HHVdPTLQsLc ~Math Python
Video Solution (EASY TO UNDERSTAND📈)
https://www.youtube.com/watch?v=A1Li_jciTZY
~CalculaCore
Video Solution by OmegaLearn
~pi_is_3.14
Video Solution
~savannahsolver
Video Solution by TheBeautyofMath
https://youtu.be/gLahuINjRzU?t=154
~IceMatrix
Video Solution by Interstigation
https://youtu.be/DvpN56Ob6Zw?t=1
~Interstigation
Video Solution by Mathematical Dexterity (50 Seconds)
https://www.youtube.com/watch?v=ScZ5VK7QTpY
Video Solution
~Education, the Study of Everything
See Also
| 2021 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America. 
