Problem Solving

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

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

What was your logic ?
Thanks.

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.

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