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# If a – b = 16 and

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If a – b = 16 and [#permalink]  21 Jun 2017, 13:41
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If a – b = 16 and $$\sqrt{a} + \sqrt{b} = 8$$ , what is the value $$\sqrt{ab}$$ ?

(A) 2
(B) 4
(C) 8
(D) 12
(E) 15
[Reveal] Spoiler: OA

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Re: If a – b = 16 and [#permalink]  25 Jul 2017, 07:59
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Carcass wrote:

If a – b = 16 and $$\sqrt{a} + \sqrt{b} = 8$$ , what is the value $$\sqrt{ab}$$ ?

(A) 2
(B) 4
(C) 8
(D) 12
(E) 15

IMPORTANT: (√a + √b)(√a - √b) = a - b [this is a cute version of difference of squares]
We're told that a - b = 16 AND √a + √b = 8
Plug these values into the equation to get: (8)(√a - √b) = 16
From this, we can see that √a - √b = 2

We have:
√a + √b = 8
√a - √b = 2

When we ADD the two equations, we get: 2√a = 10, which means √a = 5
When we SUBTRACT the bottom equation from the top, we get: 2√b = 6, which means √b = 3

So, √(ab) = (√a)(√b) = (5)(3) = 15

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Brent Hanneson – Creator of greenlighttestprep.com

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Re: If a – b = 16 and [#permalink]  25 Jul 2017, 23:45
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Hi! great question. Are there any more like this? Request you to see if its possible to post it here! Thank you
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Re: If a – b = 16 and [#permalink]  26 Jul 2017, 04:27
Expert's post
greprepclub-the-questions-vault-3523.html#p7347

More questions on their way

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Re: If a – b = 16 and [#permalink]  23 Nov 2017, 22:59
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I am posting my answer to exemplify how important it is to make sure you are taking the shortest possible route when solving a problem. I didn't do that and look at the monster I got when I tried to use substitution:

A - B = 16
A = 16 + B

Now I substituted the A for 16 + B in the other equation:

√A + √B) = 8
√(16 + B) + √B = 8

√(16 + B) = 8 - √B

Both sides are squared trying to get rid of the square root:

(√(16 + B))^2 = (8 - √B)^2
16 + B = (8 - √B) x (8 - √B)
16 + B = 64 -16√B + B
-64 -B -64 -B
-48 = -16√B

Square both sides again to get rid of the radical:

(-48)^2 = (-16√B)^2
2304 = 256B
B = 2304/256
B = 9

Now substitute the B in the first equation:

A - B = 16
A - 9 = 16
A = 25

Now plug in A and B in the third equation:

√AB = ?
√(9) x (25)
√225
= 15

Sorry it is a little confusing. This is my first post. I am a little pround I got the final result after this mathmatical ordeal.
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Re: If a – b = 16 and [#permalink]  03 Nov 2018, 12:14
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Once you hit the right road, its real easy.

a - b = (root(a) + root(b)) (root(a) - root(b))
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Re: If a – b = 16 and [#permalink]  16 Jul 2019, 03:02
Why can't we take square root of (root(a) +root(b))^2 = 8^2 and then solve for root(ab)?
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Re: If a – b = 16 and [#permalink]  16 Jul 2019, 06:20
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Why can't we take square root of (root(a) +root(b))^2 = 8^2 and then solve for root(ab)?

Can you show us what you mean?

Cheers,
Brent
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Re: If a – b = 16 and [#permalink]  17 Jul 2019, 06:43
roota+rootb = 8
(roota+rootb)^2=8^2
a+2root(a)*root(b)+b=64
2root(ab)=64-a-b

a-b=16

2root(ab)=64-16
root(ab)=48/2
root(ab)=24
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Re: If a – b = 16 and [#permalink]  17 Jul 2019, 10:15
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roota+rootb = 8
(roota+rootb)^2=8^2
a + 2(√a)(√b) + b = 64
2√(ab) = 64 - a - b

a - b = 16

2√(ab) = 64 - 16

√(ab) = 48/2
√(ab) = 24

I've highlighted a problem above.
In short, 64 - a - b is NOT the same as 64 - (a - b)

You're correct to say that: 2√(ab) = 64 - a - b
We can rewrite the right side as: 2√(ab) = 64 - (a + b)
The problem is that we don't know the value of a+b.

Does that help?

Cheers,
Brent
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Re: If a – b = 16 and [#permalink]  05 Sep 2019, 11:30
roota+rootb = 8
(roota+rootb)^2=8^2
a+2root(a)*root(b)+b=64
2root(ab)=64-a-b

a-b=16

2root(ab)=64-16
root(ab)=48/2
root(ab)=24

this is what i did too...
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Re: If a – b = 16 and [#permalink]  02 May 2020, 03:31
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I don't know if I was extremely lucky or it was intuition.

I looked at root(a) + root(b) = 8 and the a - b = 16 equations, and also the options. All are integers, no points or fractions or roots in the values. So I figured a and b have to be integers.

root(a) + root(b) = 8
This means that the roots of a and b are less than 8 because roots cannot be negative.

The first option that popped in my mind was 5 + 3 = 8. So I plugged them in the other equation and found that indeed, 25 - 9 = 16.

So root(ab) = root(a)*root(b) = 3*5 = 15.
Re: If a – b = 16 and   [#permalink] 02 May 2020, 03:31
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