Radicals: sqrt((16)(20)+ ( 8 ) (32))

II wrote:What was your logic ?

Radicals are always a problem whenever there is addition or subtraction involved.

For example: sqrt( ( 4+4) is NOT equal to sqrt (4) + sqrt (4). GMAT/GRE both test us on this common misconception.

On the other hand, if we did have multiplication or division under the radical , that makes it pretty easy.
For example: sqrt( 36 x 81) = sqrt (36) x sqrt (81).

Thats why the logic in any radical problems with addition and subtraction is to convert those mathematical operations to multiplication and division.

Factorization is one of the best tools for this.

Back to Question:

II wrote:sqrt((16)(20)+(8 )(32)) =

(A) 4 sqrt(20)
(B) 24
(C) 25
(D) 4 sqrt(20) + 8 sqrt(2)
(E) 32

sqrt((16)(20)+(8 )(32)) = ???

Since there is addition, be on the lookout for avoiding the obvious trap D

Lets factorise the part inside the radical thusly:
(16) (20) + (8 ) (32) = 16 (20) + (8 ) (2) (16)
This boils down to .......= 16{ (20) + (8 ) (2) }
which then simplifies to = (16) {20+16} = (16) (36)

This is under the radical though. Voila! You have made addition look like Multiplication.

The answer is sqrt { (16) (36)} = sqrt (16) x sqrt (36) = 4x 6 = 24

Qa = B

The above solution is 100% correct. Of course, we also could have just done some arithmetic.

sqrt((16)(20)+(8)(32)) = sqrt(320 + 256) = sqrt(576) = 24

choose (b).

Stuart Kovinsky wrote:The above solution is 100% correct. Of course, we also could have just done some arithmetic.

sqrt((16)(20)+(8)(32)) = sqrt(320 + 256) = sqrt(576) = 24

choose (b).

Just for kicks sake Stuart:

The part inside the square root is a number that ends in 6 (since 16x20 + 8x32) yields a units digit that looks like (0 +6).

That leaves only a number ending in 6 or 4 as the answer, since 6^2 = 36 and 4^2 = 16...last digit is 6.

No numbers ending in 6, so only 24 is the answer.

At this point, I am just being more silly than serious though.

sqrt((16)(20)+(Cool(32)) = What does the Cool denote?

Thanks, Stuart

"sqrt(576) = 24" is a bad answer because the average test taker does not have that memorized.

"sqrt(576) = 24" is a bad answer because the average test taker does not have that memorized.

ManifestDestiny88 - I think you are correct. The average test taker wouldn't know that the square root of 576 is 24, but who wants to be the average test taker!?

I have all my students memorize squares up to 25, powers of 2, 3, 5, and 7, fraction decimal equivalents, etc. You should never underestimate the power of math facts. Being "friends with the integers" will allow you to grind through a problem like this quickly and efficiently without worrying about the "best" way to do the problem.

The GMAT rewards knowing these facts:

• Almost all exponential growth problems are doubling problems. If the population of GMAT students doubles every year for 8 years, and you know that 2^8 = 256, then you're going to find the answer in less than 30 seconds
  
  2^10 is a good approximation for 1000
  
  When calculating by hand fractions are almost always better than decimals, so knowing your fractions decimal equivalents will simplify nasty calculations

These are just a few examples.

I've attached a pdf with all the math facts that I think are worth knowing. The list is extensive, but the payoff is worth the work.

Factor to the common variables

I like your answer. Nice and simple.

Musiq wrote:

Stuart Kovinsky wrote:The above solution is 100% correct. Of course, we also could have just done some arithmetic.

sqrt((16)(20)+(8)(32)) = sqrt(320 + 256) = sqrt(576) = 24

choose (b).

Just for kicks sake Stuart:

The part inside the square root is a number that ends in 6 (since 16x20 + 8x32) yields a units digit that looks like (0 +6).

That leaves only a number ending in 6 or 4 as the answer, since 6^2 = 36 and 4^2 = 16...last digit is 6.

No numbers ending in 6, so only 24 is the answer.

At this point, I am just being more silly than serious though.

Rewriting as products of prime factors we get:
(16)(20)+ ( 8 ) (32) = (2x2x2x2)(2x2x5) + (2x2x2)(2x2x2x2x2)
= (2^6 x 5) + (2^6 x 4) = 2^6 x 9
SQRT(2^6 x 9) = 2^3 x 3 = 8 x 3 = 24