Difference between revisions of "2021 AMC 10B Problems/Problem 2"

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~~Problem~~
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==Problem==
What is the value of <cmath>\sqrt{(3-2\sqrt{2})^2}+\sqrt{(3+2\sqrt{2})^2}?</cmath>
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What is the value of <cmath>\sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}?</cmath>
  
 
<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~~
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==Solution==
 
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Note that the square root of a squared number is the absolute value of the number.
We have that <math>\sqrt{(3-2\sqrt{2})^2} = 3-2\sqrt{2}</math> and <math>\sqrt{(3+2\sqrt{2})^2} = 3+2\sqrt{2}</math>. Adding them together, we get that it is equal to <math>\boxed{6}</math>. ~ ArduinoRasp34567
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So the first term equals <math>2\sqrt{3}-3</math> and the second term is <math>3+2\sqrt3</math>
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Summed up you get <math>\boxed{\textbf{(D)} ~4\sqrt{3}}</math>~bjc

Revision as of 18:29, 11 February 2021

Problem

What is the value of \[\sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}?\]

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~4\sqrt{3}-6 \qquad\textbf{(C)} ~6 \qquad\textbf{(D)} ~4\sqrt{3} \qquad\textbf{(E)} ~4\sqrt{3}+6$

Solution

Note that the square root of a squared number is the absolute value of the number. So the first term equals $2\sqrt{3}-3$ and the second term is $3+2\sqrt3$ Summed up you get $\boxed{\textbf{(D)} ~4\sqrt{3}}$~bjc

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