Square root /w fraction and exponent: how to simplilfy correctly

Quasimoto

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Joined
Jun 13, 2019
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6
Dear forum users,


I've been trying to figure this out.
15604635019921445585783.jpg

Could someone explain to me how this is the answer?

15604637110941456507112.jpg

Thank you in advance!
 

tkhunny

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Apr 12, 2005
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\(\displaystyle \left(\dfrac{2\sqrt{3}}{3\sqrt{2}}\right)^{3} = \left(\dfrac{2}{3}\right)^{3}\cdot\left(\dfrac{\sqrt{3}}{\sqrt{2}}\right)^{3} = \left(\dfrac{2}{3}\right)^{3}\cdot\left(\dfrac{\sqrt{3}}{\sqrt{2}}\right)^{2}\cdot\left(\dfrac{\sqrt{3}}{\sqrt{2}}\right)\)

Follow?
 

topsquark

Full Member
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Aug 27, 2012
Messages
520
1st expression = ~.5443
2nd expression = ~2.0528
As noted by the expansion that tkhunny is using the "solution" given in the OP is incorrect. The actual answer has a \(\displaystyle \sqrt{2}\) in the denominator. The fraction in front of the radicals is wrong as well.

Is this the solution to a different problem?

-Dan
 

Quasimoto

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Joined
Jun 13, 2019
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I
\(\displaystyle \left(\dfrac{2\sqrt{3}}{3\sqrt{2}}\right)^{3} = \left(\dfrac{2}{3}\right)^{3}\cdot\left(\dfrac{\sqrt{3}}{\sqrt{2}}\right)^{3} = \left(\dfrac{2}{3}\right)^{3}\cdot\left(\dfrac{\sqrt{3}}{\sqrt{2}}\right)^{2}\cdot\left(\dfrac{\sqrt{3}}{\sqrt{2}}\right)\)

Follow?
I am following it yes.
 

Quasimoto

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Jun 13, 2019
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6
As noted by the expansion that tkhunny is using the "solution" given in the OP is incorrect. The actual answer has a \(\displaystyle \sqrt{2}\) in the denominator. The fraction in front of the radicals is wrong as well.

Is this the solution to a different problem?

-Dan
I looked for the answer again and yes the solution is to a different problem.
But my original answer is still wrong.
So I will be trying to figure it out later today.

Thank you
 

pka

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Jan 29, 2005
Messages
8,717
Dear forum users, I've been trying to figure this out.
You posted two images:
\(\displaystyle \boxed{\left(\dfrac{2\sqrt3}{3\sqrt2}\right)^3}\) & \(\displaystyle \boxed{\frac{32}{27}\sqrt3}\)
The second image is not the reduction of the first.
You tell us at what step you fail to understand.
1) \(\displaystyle \frac{2\sqrt3}{3\sqrt2}=\frac{2\sqrt3\cdot\sqrt2}{3\sqrt2\cdot\sqrt2}\)
2) \(\displaystyle \frac{2\sqrt3}{3\sqrt2}=\frac{2\sqrt6}{3\cdot 2}\)
3) \(\displaystyle \frac{2\sqrt3}{3\sqrt2}=\frac{\sqrt6}{3}\)

4) \(\displaystyle \left(\frac{\sqrt6}{3}\right)^3=\frac{6\sqrt6}{27}\)
5) \(\displaystyle \boxed{\left(\frac{\sqrt6}{3}\right)^3=\frac{2\sqrt6}{9}}\) CLICK HERE
 

Quasimoto

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Joined
Jun 13, 2019
Messages
6
You posted two images:
\(\displaystyle \boxed{\left(\dfrac{2\sqrt3}{3\sqrt2}\right)^3}\) & \(\displaystyle \boxed{\frac{32}{27}\sqrt3}\)
The second image is not the reduction of the first.
You tell us at what step you fail to understand.
1) \(\displaystyle \frac{2\sqrt3}{3\sqrt2}=\frac{2\sqrt3\cdot\sqrt2}{3\sqrt2\cdot\sqrt2}\)
2) \(\displaystyle \frac{2\sqrt3}{3\sqrt2}=\frac{2\sqrt6}{3\cdot 2}\)
3) \(\displaystyle \frac{2\sqrt3}{3\sqrt2}=\frac{\sqrt6}{3}\)

4) \(\displaystyle \left(\frac{\sqrt6}{3}\right)^3=\frac{6\sqrt6}{27}\)
5) \(\displaystyle \boxed{\left(\frac{\sqrt6}{3}\right)^3=\frac{2\sqrt6}{9}}\) CLICK HERE
The second image is indeed the wrong solution to the problem.
Thank you for explaining step by step.

I failed at step 4, I used the exponent at the beginning instead of solving everything what is inside of the parentheses.

Thank you very much I appreciate the time and effort.
 
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