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

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by II » Sat Apr 12, 2008 2:45 pm
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.
Last edited by II on Mon May 05, 2008 1:38 am, edited 1 time in total.
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Re: Number Properties (Exponents + Roots)

by Musiq » Sat Apr 12, 2008 4:11 pm
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
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by Stuart@KaplanGMAT » Sat Apr 12, 2008 7:03 pm
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).
Last edited by Stuart@KaplanGMAT on Sun Apr 13, 2008 7:38 am, edited 1 time in total.
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by Musiq » Sat Apr 12, 2008 9:12 pm
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. :D
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by gmat765 » Wed Apr 16, 2008 7:00 pm
sqrt((16)(20)+(Cool(32)) = What does the Cool denote?
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by hillzheng » Wed May 14, 2008 4:51 pm
Thanks, Stuart
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by ManifestDestiny88 » Fri May 29, 2015 1:15 pm
"sqrt(576) = 24" is a bad answer because the average test taker does not have that memorized.
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by Brent@GMATPrepNow » Fri May 29, 2015 1:37 pm
sqrt[(16)(20)+(8)(32)]
One option here is to evaluate (16)(20)+(8)(32), and then find the square root of the result. That's a bit of work.

We can also apply a technique called "Multiplying by Doubling and Halving"
We have a free video on this: https://www.gmatprepnow.com/module/gener ... es?id=1113

In the first part, (16)(20), I notice that 16 is a perfect square. Nice!
In the second part, (8)(32), I notice that we have no perfect squares. However, using the doubling and halving technique, we can see that (8)(32) = (16)(16)

So, sqrt[(16)(20)+(8)(32)] = sqrt[(16)(20)+(16)(16)]
= sqrt[16(20 + 16)] {I factored out the 16}
= sqrt[(16)(36)]

At this point, we can apply a useful rule: sqrt(xy) = [sqrt(x)][sqrt(y)]

sqrt[(16)(36)] = [sqrt(16)][sqrt(36)]
= (4)(6)
= 24

Cheers,
Brent
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by John@GMATPrepNow » Sat May 30, 2015 12:56 pm
"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.
Attachments
Math Facts Summary.pdf
(503.23 KiB) Downloaded 209 times
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by nikhilgmat31 » Mon Jun 01, 2015 11:25 pm
Factor to the common variables :)
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by Mathsbuddy » Tue Jun 02, 2015 7:10 am
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. :D
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by Mathsbuddy » Tue Jun 02, 2015 7:18 am
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
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