Problem Solving

Max@Math Revolution wrote:(√27 + √243)/√54=?
A. 2√2
B. 2√3
C. 3√2
D. 3√3
E. √2

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) = [spoiler]2/√2 = A[/spoiler]

RELATED RESOURCES
For more information on simplifying roots, watch these videos
- https://www.gmatprepnow.com/module/gmat ... video/1038
- https://www.gmatprepnow.com/module/gmat ... video/1044

Cheers,
Brent

nasahtahir wrote:Brent would you kindly explain if we take 4/√2 and turn it into 2^2/√2= 2^2/ 2^1/2 we have a common base do we not? We are allowed to subtract the powers at will? Yes? Or do I have it all wrong?
appreciate your help.

HT

Sure, we can think of 4/√2 as being equal to (2^2)/[2^(1/2)]
Then, because we have the same base, we can subtract the exponents (2 - 1/2) to get 2^(3/2).
From here, 2^(3/2) = (2^3)^(1/2) = 8^(1/2) = √8 = 2√2
So, your idea works. It just requires us to know how to deal with fractional exponents.


Alternatively, what I did was create a fraction that's equivalent to 4/√2
So, I took 4/√2 and multiplied numerator and denominator by √2 to get:4√2/(√2)(√2)
This simplifies to 4√2/2, which equals 2√2

Does that help?

Cheers,
Brent

Max@Math Revolution 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