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(27 + 243)/54 = ?

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MathRevolution
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(27 + 243)/54 = ? [#permalink] New post   01 Jun 2016, 06:47
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\(\frac{√27+√243}{√54}=?\)

A. 2√2
B. 2√3
C. 3√2
D. 3√3
E. √2

 
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BrentGMATPrepNow
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Re: (27 + 243)/54 = ? [#permalink] New post   Updated on: 30 Jul 2022, 08:11
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MathRevolution wrote:
(√27+√243)/√54=?
A. 2√2
B. 2√3
C. 3√2
D. 3√3
E. √2

*An answer will be posted in 2 days.


We'll use the following rule: √(xy) = (√x)(√y)

√27 = √[(9)(3)] = (√9)(√3) = 3√3
√243 = √[(81)(3)] = (√81)(√3) = 9√3
√54 = √[(9)(6)] = (√9)(√6) = 3√6

So, (√27 + √243)/√54 = (3√3 + 9√3)/(3√6)
= (12√3)/(3√6)
= 4/√2
Check answer choices.... not there.

Take 4/√2 and multiply top and bottom by √2.
We get: (4/√2)(√2/√2) = (4√2)/(2) = 2√2 = A

Cheers,
Brent

Originally posted by BrentGMATPrepNow on 01 Jun 2016, 12:12.
Last edited by BrentGMATPrepNow on 30 Jul 2022, 08:11, edited 4 times in total.
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Re: (27 + 243)/54 = ? [#permalink] New post   01 Jun 2016, 12:15
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MathRevolution wrote:
(√27+√243)/√54=?
A. 2√2
B. 2√3
C. 3√2
D. 3√3
E. √2

*An answer will be posted in 2 days.


\(\sqrt{27}\) = \(3\sqrt{3}\)

\(\sqrt{243}\) = \(9\sqrt{3}\)

\(\sqrt{54}\) = \(3\sqrt{3}\sqrt{2}\)


\(\sqrt{27}\) + \(\sqrt{243}\) = \(3\sqrt{3}\) + \(9\sqrt{3}\)

\(\sqrt{27}\) + \(\sqrt{243}\) = \(12\sqrt{3}\)

( \(\sqrt{27}\) + \(\sqrt{243}\) ) / \(\sqrt{54}\) = ( \(12\sqrt{3}\) ) / ( \(3\sqrt{3}\sqrt{2}\) )


\(3\sqrt{3}\sqrt{2}\) = \(4/\sqrt{2}\) = \(2\sqrt{2}\)

Hence answer will be (A)
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Re: (27 + 243)/54 = ? [#permalink] New post   05 Jun 2016, 05:48
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(√27+√243)/√54=(3√3+9√3)/3√6=12√3/3√6=4/√2=2√2. Hence, the correct answer is A.
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Re: (27 + 243)/54 = ? [#permalink] New post   05 Jun 2016, 06:17
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Once you notice the following, figuring out the exact value of each of the given values can be eliminated and that should save some time :)

\(\frac{\sqrt{27} + \sqrt{243}}{\sqrt{54}} = \frac{ \sqrt{27} + ( \sqrt{27} \times \sqrt{9} ) } { \sqrt{27} \times \sqrt{2} } = \frac{ \sqrt{27} ( 1 + \sqrt{9} ) } { \sqrt{27} \times \sqrt{2} } = \frac{4}{\sqrt{2}} = 2\sqrt{2}\)
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Re: (27 + 243)/54 = ? [#permalink] New post   24 Aug 2016, 20:57
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I still don't see how 12√3)/(3√6) gives 2√2
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Re: (27 + 243)/54 = ? [#permalink] New post   23 Nov 2016, 07:33
cicerone wrote:
Once you notice the following, figuring out the exact value of each of the given values can be eliminated and that should save some time :)

\(\frac{\sqrt{27} + \sqrt{243}}{\sqrt{54}} = \frac{ \sqrt{27} + ( \sqrt{27} \times \sqrt{9} ) } { \sqrt{27} \times \sqrt{2} } = \frac{ \sqrt{27} ( 1 + \sqrt{9} ) } { \sqrt{27} \times \sqrt{2} } = \frac{4}{\sqrt{2}} = 2\sqrt{2}\)

Could you or somebody please explain how 4 divided by square root of 2 equals 2 times square root of 2? Not sure why but I don't get it.
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Re: (27 + 243)/54 = ? [#permalink] New post   23 Nov 2016, 07:51
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gracie90 wrote:
cicerone wrote:
Once you notice the following, figuring out the exact value of each of the given values can be eliminated and that should save some time :)

\(\frac{\sqrt{27} + \sqrt{243}}{\sqrt{54}} = \frac{ \sqrt{27} + ( \sqrt{27} \times \sqrt{9} ) } { \sqrt{27} \times \sqrt{2} } = \frac{ \sqrt{27} ( 1 + \sqrt{9} ) } { \sqrt{27} \times \sqrt{2} } = \frac{4}{\sqrt{2}} = 2\sqrt{2}\)

Could you or somebody please explain how 4 divided by square root of 2 equals 2 times square root of 2? Not sure why but I don't get it.


Multiply \(\frac{4}{\sqrt{2}}\) by \(\frac{\sqrt{2}}{\sqrt{2}}\) to get \(\frac{4}{\sqrt{2}}*\frac{\sqrt{2}}{\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\)

This algebraic manipulation is called rationalization and is performed to eliminate irrational expression in the denominator.

Questions involving rationalization to practice:
if-x-0-then-106291.html
if-n-is-positive-which-of-the-following-is-equal-to-31236.html
consider-a-quarter-of-a-circle-of-radius-16-let-r-be-the-131083.html
in-the-diagram-not-drawn-to-scale-sector-pq-is-a-quarter-139282.html
in-the-diagram-what-is-the-value-of-x-129962.html
the-perimeter-of-a-right-isoscles-triangle-is-127049.html
which-of-the-following-is-equal-to-98531.html
if-x-is-positive-then-1-root-x-1-root-x-163491.html
1-2-sqrt3-64378.html
if-a-square-mirror-has-a-20-inch-diagonal-what-is-the-99359.html

Hope it helps.
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Re: (27 + 243)/54 = ? [#permalink] New post   08 Dec 2017, 11:34
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MathRevolution wrote:
(√27+√243)/√54=?
A. 2√2
B. 2√3
C. 3√2
D. 3√3
E. √2


We are given (√27+√243)/√54. Let’s simplify each term.

√27 = √9 x √3 = 3√3

√243 = √81 x √3 = 9√3

√54 = √9 x √6 = 3√6

Thus:

(√27+√243)/√54 = (3√3 + 9√3)/(3√6) = (12√3)/(3√6) = 4/√2

Finally, we can rationalize 4/√2 by multiplying the numerator and denominator by √2 and we obtain:

(4/√2)(√2/√2) = (4√2)/2 = 2√2

Answer: A
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Re: (27 + 243)/54 = ? [#permalink] New post   24 Feb 2019, 15:26
How to know when does it matter rationalize and then does it not?
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Re: (27 + 243)/54 = ? [#permalink] New post   24 Oct 2019, 06:51
MathRevolution wrote:
(√27+√243)/√54=?
A. 2√2
B. 2√3
C. 3√2
D. 3√3
E. √2

*An answer will be posted in 2 days.


This is how i solved

√27+√243)/√54 =

First take out the factors for √27 , √243 and √ 54

Eg factors for √27 = √ 3*3*3 = 3√3 similarly for √ 243 = 9√3 and for √54 = 3√6

= ( 3√3 +9√3) / 3√6 = 12√3 / 3√6 = 4/√2 ,,,,, now check ans choice but we dont have the ans choice in this form so futher simplify'

= 2*2/√2 and we also know that √2* √2 =2 so lets write one of the 2 in √ form

= √2*√2 *2 / √2 cancel √2 from numerator and denoiminator to get 2√2 = ans choice A
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Re: (27 + 243)/54 = ? [#permalink] New post   11 Jun 2023, 04:31
MathRevolution wrote:
(√27+√243)/√54=?
A. 2√2
B. 2√3
C. 3√2
D. 3√3
E. √2

*An answer will be posted in 2 days.


As simple as these questions seem to be, they always make my head :dazed . Personally I don't like to see fractions as powers so there's a slightly alternate way that I follow which makes it seem way less complicated(at least for me :-P ) Hope others like it :

(√27+√243)/√54
=> 27^1/2 + 243^1/2 / (3^3 * 2)^1/2
=> 3^3/2 + 3^5/2 / 3^3/2 * √2

=> Substituting √3 = A...
=> A^3 + A^5 / A^3 * √2 ....(Now brain is working again)
=> A^3 * (1 + A^2) / A^3 * √2 Taking A^3 as common
=> (1 + A^2) / √2
=> 1+ (√3)^2 / √2 ....(Substituting back)
=> 4/√2 => 2√2
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Re: (27 + 243)/54 = ? [#permalink]
11 Jun 2023, 04:31
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