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(√27+√243)/√54=?

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(√27+√243)/√54=? [#permalink]

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(√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.
[Reveal] Spoiler: OA

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(√27+√243)/√54=? [#permalink]

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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

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Last edited by GMATPrepNow on 23 Nov 2016, 06:54, edited 2 times in total.

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Re: (√27+√243)/√54=? [#permalink]

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New post 01 Jun 2016, 11:15
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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]

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New post 05 Jun 2016, 04:48
(√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]

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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]

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New post 24 Aug 2016, 19:57
I still don't see how 12√3)/(3√6) gives 2√2

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Re: (√27+√243)/√54=? [#permalink]

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New post 23 Nov 2016, 06:33
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]

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New post 23 Nov 2016, 06:51
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]

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New post 08 Dec 2017, 10:34
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] 08 Dec 2017, 10:34
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