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2021 AMC 10B Problems/Problem 2

Problem

What is the value of $\sqrt{\left(3-2\sqrt{3}\right)^2}+\sqrt{\left(3+2\sqrt{3}\right)^2}$?

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

Solution 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 $3$ and $2\sqrt{3}$, we see that $2\sqrt{3}>3$, thus $3-2\sqrt{3}$ is negative, so we must take the absolute value of $3-2\sqrt{3}$, which is just $2\sqrt{3}-3$. Knowing this, the first term in the expression equals $2\sqrt{3}-3$ and the second term is $3+2\sqrt3$, and summing the two gives $\boxed{\textbf{(D)} ~4\sqrt{3}}$.

~bjc, abhinavg0627 and JackBocresion

Solution 2

Let $x = \sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}$, then $x^2 = (3-2\sqrt{3})^2+2\sqrt{(-3)^2}+(3+2\sqrt3)^2$. The $2\sqrt{(-3)^2}$ term is there due to difference of squares when you simplify $2ab$ from $(a + b)^2$. Simplifying the expression gives us $x^2 = 48$, so $x=\boxed{\textbf{(D)} ~4\sqrt{3}}$ ~ 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

\[\sqrt{(3-2 \cdot 1.7)^2} + \sqrt{(3+2 \cdot 1.7)^2}\]

\[= \sqrt{(-0.4)^2} + \sqrt{(6.4)^2}\]

\[= \sqrt{0.16} + \sqrt{6.4^2}\]

\[= 0.4 + 6.4\]

\[= 6.8\]

Looking at the answer choices, we see that $\boxed{\textbf{(D)} ~4\sqrt{3}}$ gives $6.8$ 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

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, as part of the American Mathematics Competitions. AMC Logo.png

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