Oct 16 08:00 PM PDT  09:00 PM PDT EMPOWERgmat is giving away the complete Official GMAT Exam Pack collection worth $100 with the 3 Month Pack ($299) Oct 18 08:00 AM PDT  09:00 AM PDT Learn an intuitive, systematic approach that will maximize your success on Fillintheblank GMAT CR Questions. Oct 19 07:00 AM PDT  09:00 AM PDT Does GMAT RC seem like an uphill battle? eGMAT is conducting a free webinar to help you learn reading strategies that can enable you to solve 700+ level RC questions with at least 90% accuracy in less than 10 days. Sat., Oct 19th at 7 am PDT Oct 20 07:00 AM PDT  09:00 AM PDT Get personalized insights on how to achieve your Target Quant Score. Oct 22 08:00 PM PDT  09:00 PM PDT On Demand for $79. For a score of 4951 (from current actual score of 40+) AllInOne Standard & 700+ Level Questions (150 questions)
Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 58381

If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
06 Mar 2017, 01:49
Question Stats:
68% (02:18) correct 32% (02:34) wrong based on 424 sessions
HideShow timer Statistics
If \(4^a∗2^b=16^4\), what is the minimum possible value of (32–ab)? A. 32 B. 16 C. 8 D. 0 E. 16
Official Answer and Stats are available only to registered users. Register/ Login.
_________________




Retired Moderator
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1333
Location: Viet Nam

Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
07 Mar 2017, 21:39
Bunuel wrote: If \(4^a∗2^b=16^4\), what is the minimum possible value of (32–ab)?
A. 32 B. 16 C. 8 D. 0 E. 16 \(4^a * 2^b = 16^4 \iff 2^{2a} * 2^b = 2 ^ 16 \iff 2a+b=16 \) If either \(a\) or \(b\) is negative, then \(ab\) is negative. Hence \(32ab > 32\). If either \(a\) or \(b\) equals to 0, then \(32ab=32\) If both \(a\) and \(b\) is positive, using AMGM inequality, we have \(16=2a+b \geq 2 \sqrt{2a*b} \implies 8 \geq \sqrt{2ab} \implies ab \leq 32\). \(ab=32 \iff 2a=b=8 \iff a=4\) and \(b=8\). Hence, \(32  ab \geq 3232=0\). The answer is D
_________________




Manager
Joined: 02 Aug 2015
Posts: 153

Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
06 Mar 2017, 07:51
Bunuel wrote: If \(4^a∗2^b=16^4\), what is the minimum possible value of (32–ab)?
A. 32 B. 16 C. 8 D. 0 E. 16 \(4^a∗2^b=16^4\) > 2a+b=16. When a=4 and b=8, ab=32 hence the lowest value of (32ab) is 0. Hence I'm getting D.



Veritas Prep GMAT Instructor
Affiliations: Veritas Prep
Joined: 21 Dec 2014
Posts: 45
Location: United States (DC)
GPA: 3.11
WE: Education (Education)

Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
07 Mar 2017, 21:13
Diwakar003 wrote: Bunuel wrote: If \(4^a∗2^b=16^4\), what is the minimum possible value of (32–ab)?
A. 32 B. 16 C. 8 D. 0 E. 16 \(4^a∗2^b=16^4\) > 2a+b=16. When a=4 and b=8, ab=32 hence the lowest value of (32ab) is 0. Hence I'm getting D. Okay, but how can you be absolutely sure that that's the absolute lowest possible?



Veritas Prep GMAT Instructor
Affiliations: Veritas Prep
Joined: 21 Dec 2014
Posts: 45
Location: United States (DC)
GPA: 3.11
WE: Education (Education)

Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
07 Mar 2017, 21:43
nguyendinhtuong wrote: Bunuel wrote: If \(4^a∗2^b=16^4\), what is the minimum possible value of (32–ab)?
A. 32 B. 16 C. 8 D. 0 E. 16 \(4^a * 2^b = 16^4 \iff 2^{2a} * 2^b = 2 ^ 16 \iff 2a+b=16 \) If either \(a\) or \(b\) is negative, then \(ab\) is negative. Hence \(32ab > 32\). If either \(a\) or \(b\) equals to 0, then \(32ab=32\) If both \(a\) and \(b\) is positive, using AMGM inequality, we have \(16=2a+b \geq 2 \sqrt{2a*b} \implies 8 \geq \sqrt{2ab} \implies ab \leq 32\). \(ab=32 \iff 2a=b=8 \iff a=4\) and \(b=8\). Hence, \(32  ab \geq 3232=0\). The answer is D Fancy. Note that AMGM is pretty firmly beyond the scope of necessary GMAT tools, but this does work. However, also note that there is a way to prove the minimum without resorting to this tool...



Director
Joined: 21 Mar 2016
Posts: 509

Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
07 Mar 2017, 23:32
AnthonyRitz wrote: nguyendinhtuong wrote: Bunuel wrote: If \(4^a∗2^b=16^4\), what is the minimum possible value of (32–ab)?
A. 32 B. 16 C. 8 D. 0 E. 16 \(4^a * 2^b = 16^4 \iff 2^{2a} * 2^b = 2 ^ 16 \iff 2a+b=16 \) If either \(a\) or \(b\) is negative, then \(ab\) is negative. Hence \(32ab > 32\). If either \(a\) or \(b\) equals to 0, then \(32ab=32\) If both \(a\) and \(b\) is positive, using AMGM inequality, we have \(16=2a+b \geq 2 \sqrt{2a*b} \implies 8 \geq \sqrt{2ab} \implies ab \leq 32\). \(ab=32 \iff 2a=b=8 \iff a=4\) and \(b=8\). Hence, \(32  ab \geq 3232=0\). The answer is D Fancy. Note that AMGM is pretty firmly beyond the scope of necessary GMAT tools, but this does work. However, also note that there is a way to prove the minimum without resorting to this tool... AnthonyRitz please explain another way



Veritas Prep GMAT Instructor
Affiliations: Veritas Prep
Joined: 21 Dec 2014
Posts: 45
Location: United States (DC)
GPA: 3.11
WE: Education (Education)

If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
Updated on: 07 Mar 2017, 23:54
mohshu wrote: AnthonyRitz please explain another way Fair enough. Try this: \(2a+b=16\) \(32ab = 32a(162a) = 3216a+2a^2 = 2(a^28a+16) = 2(a4)^2\) Since \((a4)^2 \geq 0\) with equality when \(a = 4\), the minimum is \(0\) when \(a = 4\) and \(b = 8\).
Originally posted by AnthonyRitz on 07 Mar 2017, 23:39.
Last edited by AnthonyRitz on 07 Mar 2017, 23:54, edited 2 times in total.



Retired Moderator
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1333
Location: Viet Nam

If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
07 Mar 2017, 23:46
AnthonyRitz wrote: nguyendinhtuong wrote: Bunuel wrote: If \(4^a∗2^b=16^4\), what is the minimum possible value of (32–ab)?
A. 32 B. 16 C. 8 D. 0 E. 16 \(4^a * 2^b = 16^4 \iff 2^{2a} * 2^b = 2 ^ 16 \iff 2a+b=16 \) If either \(a\) or \(b\) is negative, then \(ab\) is negative. Hence \(32ab > 32\). If either \(a\) or \(b\) equals to 0, then \(32ab=32\) If both \(a\) and \(b\) is positive, using AMGM inequality, we have \(16=2a+b \geq 2 \sqrt{2a*b} \implies 8 \geq \sqrt{2ab} \implies ab \leq 32\). \(ab=32 \iff 2a=b=8 \iff a=4\) and \(b=8\). Hence, \(32  ab \geq 3232=0\). The answer is D Fancy. Note that AMGM is pretty firmly beyond the scope of necessary GMAT tools, but this does work. However, also note that there is a way to prove the minimum without resorting to this tool... There is another way. \((2a)+b=16 \implies ((2a)+b)^2=16^2 \implies (2a)^2 + b^2 + 2*(2a)*b=16^2\) \((2a)^2 + b^2  2*(2a)*b = 16^2  4*(2a)*b \implies (2ab)^2 = 16^2  4*(2a)*b\) We have \((2ab)^2 \geq 0 \;\; \forall a,b \in R\). Hence \(16^2 4 *(2a)*b \geq 0 \implies 4*(2a)*b \leq 16^2 \implies 8ab \leq 16^2 \implies ab\leq 32\)
_________________



Director
Status: Come! Fall in Love with Learning!
Joined: 05 Jan 2017
Posts: 532
Location: India

Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
08 Mar 2017, 01:58
LHS: 4^a x 2^b = 2^(2a+b) RHS: 16^4 = 2^16 hence 2a+b =16 or b = 162a putting in the expression 32ab we get 32a(162a) = 3216a +2a^2 minima of quadratic equation comes at 16/4 = 4. for a = 4 b = 8. so ab = 32.hence 32ab = 3232 = 0. Option D
_________________
GMAT Mentors



Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8069
Location: United States (CA)

Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
09 Mar 2017, 17:15
Bunuel wrote: If \(4^a∗2^b=16^4\), what is the minimum possible value of (32–ab)?
A. 32 B. 16 C. 8 D. 0 E. 16 We can simplify the given equation by reexpressing 4 as 2^2 and 16 as 2^4: (2^2)^a * 2^b = (2^4)^4 2^(2a) * 2^b = 2^16 2^(2a + b) = 2^16 In an equation, when the bases are the same, the exponents must be equal: 2a + b = 16 b = 16  2a We need to find the minimum value of (32  ab). We have determined that b = 16  2a, so we need to find the minimum value of 32  a(16  2a). Let’s simplify this expression: 32  16a + 2a^2 2a^2  16a + 32 The above is a quadratic expression. Recall that the graph of y = ax^2 + bx + c is a parabola. It opens up when a > 0, and its vertex will be the minimum point. To find the xvalue of the vertex, we can use the formula x = b/(2a). As for the minimum value of the quadratic expression (i.e., the y value), we can plug the xvalue of the vertex back into the expression. Thus, the minimum value of the expression occurs when a = (16)/[2(2)] = 16/4 = 4, and the minimum value is: 2(4)^2  16(4) + 32 32  64 + 32 = 0 Answer: D
_________________
5star rated online GMAT quant self study course See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews If you find one of my posts helpful, please take a moment to click on the "Kudos" button.



Manager
Joined: 08 Oct 2016
Posts: 201
Location: United States
Concentration: General Management, Finance
GPA: 2.9
WE: Engineering (Telecommunications)

Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
20 Apr 2017, 13:39
Took a little more time but solved it Choice is: D here you go 4^a*2^b=16^4 2^2a*2^b=2^16 so 2a+b=16 option E ruled out min we get is Zero So check 2(4)+8=16 LHS=RHS put in 32ab 32(4)(8) 0 Here is our answer
_________________
Got Q42,V17 Target#01 Q45,V20April End



Intern
Joined: 03 Jul 2018
Posts: 4

Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
11 Sep 2018, 09:33
Well, once you have the equation 2a+b=16, equate answer choices to "32ab". I tried with choice D  and it satisfied both eqns. The only other answer choice lower than 0 was E, which does not satisfy 2a+b=16. So we're left with D.



Intern
Joined: 03 Sep 2018
Posts: 7

Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
17 Sep 2018, 07:58
I think I went about this the wrong way but I ended up with the right answer:
\(4^a * 2^b = 16^4\) \(4^a * 2^b = (2 * 2 * 4)^4\) \(4^a * 2^b = (2^2 * 4)^4\) \(4^a * 2^b = 2^8 * 4^4\)
So I took \(a = 4\) and \(b = 8\) and plugged it into \((32  ab)\) and got 0. D. I didn't feel totally confident about it though.



Intern
Joined: 09 May 2016
Posts: 41

Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
Show Tags
23 Sep 2018, 03:32
broall wrote: Bunuel wrote: If \(4^a∗2^b=16^4\), what is the minimum possible value of (32–ab)?
A. 32 B. 16 C. 8 D. 0 E. 16 \(4^a * 2^b = 16^4 \iff 2^{2a} * 2^b = 2 ^ 16 \iff 2a+b=16 \) If either \(a\) or \(b\) is negative, then \(ab\) is negative. Hence \(32ab > 32\). If either \(a\) or \(b\) equals to 0, then \(32ab=32\) If both \(a\) and \(b\) is positive, using AMGM inequality, we have \(16=2a+b \geq 2 \sqrt{2a*b} \implies 8 \geq \sqrt{2ab} \implies ab \leq 32\). \(ab=32 \iff 2a=b=8 \iff a=4\) and \(b=8\). Hence, \(32  ab \geq 3232=0\). The answer is D Can someone pls explain the AMGM theory here. Thanks a ton in advance




Re: If 4^a∗2^b=16^4, what is the minimum possible value of (32–ab)?
[#permalink]
23 Sep 2018, 03:32






