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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GMAT OG 13th edition - #35
Source: Beat The GMAT — Problem Solving |
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You could also do a little fun prime factorization.
16 = 2^4
20 = 2^2 * 5
8= 2^3
32 = 2^5
So we want sqrt(2^4 * 2^2 * 5 + 2^3 * 2^5)
simplified: sqrt(2^6 * 5 + 2^8)
Factor out a 2^6 to get: sqrt(2^6(5 + 2^2))
sqrt(2^6(5 + 2^2)) = sqrt(2^6 * 9) = sqrt(2^6 * 3^2) = [2^6 * 3^2]^(1/2) = 2^3 * 3 = 8 * 3 = 24
16 = 2^4
20 = 2^2 * 5
8= 2^3
32 = 2^5
So we want sqrt(2^4 * 2^2 * 5 + 2^3 * 2^5)
simplified: sqrt(2^6 * 5 + 2^8)
Factor out a 2^6 to get: sqrt(2^6(5 + 2^2))
sqrt(2^6(5 + 2^2)) = sqrt(2^6 * 9) = sqrt(2^6 * 3^2) = [2^6 * 3^2]^(1/2) = 2^3 * 3 = 8 * 3 = 24
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Hi lucas211,
When it comes to simplifying/solving these types of radicals, you are NOT allowed to "split" the calculation, but the math itself isn't that bad...
Here, you have a straight-forward enough calculation:
16(20) + 8(32)
320 + 256
576
From here, you can either take the square root of 576 OR... since the answer choices are all NUMBERS, you can use the answer choices and square them until you find one that equals 576.
Either way, you'll get 24
GMAT assassins aren't born, they're made,
Rich
When it comes to simplifying/solving these types of radicals, you are NOT allowed to "split" the calculation, but the math itself isn't that bad...
Here, you have a straight-forward enough calculation:
16(20) + 8(32)
320 + 256
576
From here, you can either take the square root of 576 OR... since the answer choices are all NUMBERS, you can use the answer choices and square them until you find one that equals 576.
Either way, you'll get 24
GMAT assassins aren't born, they're made,
Rich
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The first thing when you see such questions should be to simplifylucas211 wrote:Hello BTG
would appreciate a little help on the following question:
Thanks in advance
16*20 + 8*32 = 320 + 256 = 576
Now, (576)^1/2 = 24
Correct Option: B
This is actually one of my favorite problems in all of the OGs! There's a lot to learn about good PS process here.
I would *NOT* recommend calculating here. I think it's unrealistic to think that someone without a solid quant background would execute that flawlessly in 2 minutes. You are definitely NOT expected to know the square root of 576 off the top of your head, without a calculator!
Brent's solution of factoring out a 16 or David's solution of doing prime factorization are the best approaches here.
But... there is something you can and should do before you start solving, though. With PS questions, you should always scan the answer choices first, and see if you can eliminate any that are nonsensical or obvious traps.
The problem gives us a single root sign with a complicated sum underneath. We know that we're never allowed to split the root of a sum into the sum of roots:
So, we can automatically eliminate D. There's no way to get a sum of 2 roots from the root of one sum.
We can also eliminate A, since it's unsimplified. sqrt(20) can be simplified to 2(sqrt(5)). The GMAT will never have an unsimplified root in a correct answer. (You may also have noticed that A would just represent the first half of the sqrt in the problem - another reason to eliminate it).
Now that we're left with B, C, and D, we know that the right answer must be an integer. This implies that what's underneath the root sign must be a perfect square.
The square of 25 would have to be an odd integer, ending in 5. If all of the terms under the root sign are even, there's no way we'll end up with an odd result. Eliminate C.
Now, looking at B and E, we can get the right answer with very little effort. 32 is a number in the problem, making it less likely to be right. But also, when we square 32, we'll get some number with a units digit of 4. When we square 24, we'll get some number with a units digit of 6.
See how quickly you can find the units digit of what's under the root sign: 6*0 = 0, 8*2=6, 0 + 6 = 6. We need a units digit of 6.
The answer must be B.
I would *NOT* recommend calculating here. I think it's unrealistic to think that someone without a solid quant background would execute that flawlessly in 2 minutes. You are definitely NOT expected to know the square root of 576 off the top of your head, without a calculator!
Brent's solution of factoring out a 16 or David's solution of doing prime factorization are the best approaches here.
But... there is something you can and should do before you start solving, though. With PS questions, you should always scan the answer choices first, and see if you can eliminate any that are nonsensical or obvious traps.
The problem gives us a single root sign with a complicated sum underneath. We know that we're never allowed to split the root of a sum into the sum of roots:
So, we can automatically eliminate D. There's no way to get a sum of 2 roots from the root of one sum.
We can also eliminate A, since it's unsimplified. sqrt(20) can be simplified to 2(sqrt(5)). The GMAT will never have an unsimplified root in a correct answer. (You may also have noticed that A would just represent the first half of the sqrt in the problem - another reason to eliminate it).
Now that we're left with B, C, and D, we know that the right answer must be an integer. This implies that what's underneath the root sign must be a perfect square.
The square of 25 would have to be an odd integer, ending in 5. If all of the terms under the root sign are even, there's no way we'll end up with an odd result. Eliminate C.
Now, looking at B and E, we can get the right answer with very little effort. 32 is a number in the problem, making it less likely to be right. But also, when we square 32, we'll get some number with a units digit of 4. When we square 24, we'll get some number with a units digit of 6.
See how quickly you can find the units digit of what's under the root sign: 6*0 = 0, 8*2=6, 0 + 6 = 6. We need a units digit of 6.
The answer must be B.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
Another idea is to look for a nice root:
√(16*20 + 8*32) =>
√(64*5 + 64*4) =>
√(64*(5+4)) =>
√(64*9) => √64 * √9 => 24
√(16*20 + 8*32) =>
√(64*5 + 64*4) =>
√(64*(5+4)) =>
√(64*9) => √64 * √9 => 24
Hi Ceilidh.ericksenceilidh.erickson wrote:This is actually one of my favorite problems in all of the OGs! There's a lot to learn about good PS process here.
I would *NOT* recommend calculating here. I think it's unrealistic to think that someone without a solid quant background would execute that flawlessly in 2 minutes. You are definitely NOT expected to know the square root of 576 off the top of your head, without a calculator!
Brent's solution of factoring out a 16 or David's solution of doing prime factorization are the best approaches here.
But... there is something you can and should do before you start solving, though. With PS questions, you should always scan the answer choices first, and see if you can eliminate any that are nonsensical or obvious traps.
The problem gives us a single root sign with a complicated sum underneath. We know that we're never allowed to split the root of a sum into the sum of roots:
So, we can automatically eliminate D. There's no way to get a sum of 2 roots from the root of one sum.
We can also eliminate A, since it's unsimplified. sqrt(20) can be simplified to 2(sqrt(5)). The GMAT will never have an unsimplified root in a correct answer. (You may also have noticed that A would just represent the first half of the sqrt in the problem - another reason to eliminate it).
Now that we're left with B, C, and D, we know that the right answer must be an integer. This implies that what's underneath the root sign must be a perfect square.
The square of 25 would have to be an odd integer, ending in 5. If all of the terms under the root sign are even, there's no way we'll end up with an odd result. Eliminate C.
Now, looking at B and E, we can get the right answer with very little effort. 32 is a number in the problem, making it less likely to be right. But also, when we square 32, we'll get some number with a units digit of 4. When we square 24, we'll get some number with a units digit of 6.
See how quickly you can find the units digit of what's under the root sign: 6*0 = 0, 8*2=6, 0 + 6 = 6. We need a units digit of 6.
The answer must be B.
Thank you very much for the great explanation and also talking you time to explain your hole thought-process.
Much appreciated.
Lucas
My pleasure!Hi Ceilidh.ericksen
Thank you very much for the great explanation and also talking you time to explain your hole thought-process.
Much appreciated.
Lucas Smile
I always recommend stepping back, thinking conceptually, and trying to eliminate answers before diving in. It works on many more problems than you might think!
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
