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# Which of the following best approximates the value of q if

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Which of the following best approximates the value of q if [#permalink]  23 Aug 2010, 08:32
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Which of the following best approximates the value of q if 5^28+3^11=5^q ?

A. 39
B. 30
C. 28
D. 27
E. 17
[Reveal] Spoiler: OA

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Last edited by Bunuel on 01 Apr 2012, 23:18, edited 2 times in total.
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Re: Estimations [#permalink]  23 Aug 2010, 09:12
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Which of the following best approximates the value of q if $$5^{28}+3^{11}=5^q$$?

A. 39
B. 30
C. 28
D. 27
E. 17

Note that we need approximate value of $$q$$. Now, $$5^{28}$$ is much, much, much bigger than $$3^{11}$$. So adding $$3^{11}$$ to $$5^{28}$$ will be closer to $$5^{28}$$ then to $$5^{29}$$, basically $$3^{11}$$ is negligible in this case. So $$q\approx{28}$$.

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Re: Estimations [#permalink]  24 Aug 2010, 05:47
thx i guess i need to go review my exponents haha. i was trying to actually do the problem until it started to get out of hand....sigh...
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Which of the following best approximates the value of q if [#permalink]  01 Apr 2012, 22:40
Which of the following best approximates the value of q if 5^28 + 3^11 = 5^q ?

(A) 39
(B) 30
(C) 28
(D) 27
(E) 17

Last edited by Bunuel on 01 Apr 2012, 22:59, edited 1 time in total.
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Re: PS: Approximation [#permalink]  01 Apr 2012, 22:51
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Smita04 wrote:
Which of the following best approximates the value of q if 5^28 + 3^11 = 5^q ?
(A) 39
(B) 30
(C) 28
(D) 27
(E) 17

The answer has to be at least 28 since you are adding something to 5^28.

Note that 3^11 is much much smaller than 5^28 so adding it will not change the value of 5^28 much. If it is hard to visualize, think of it this way:

The next possible answer is 5^30 which is greater than 5^28
5^30 = 25*5^28

But
5^28 + 5^28 = 2*5^28
Even if we add 5^28 to 5^28, all we get is 2*5^28 which is much much smaller than 25*5^28
In fact, what we are actually adding is only 3^11 so 5^28 + 3^11 must be much closer to 5^28. Hence its value must be approximately 5^28.
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Re: PS: Approximation [#permalink]  01 Apr 2012, 22:56
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Smita04 wrote:
Which of the following best approximates the value of q if 5^28 + 3^11 = 5^q ?
(A) 39
(B) 30
(C) 28
(D) 27
(E) 17

Pay attention to the funda here that 3^11 is much less than 5^28 (obviously its superlarge number)

hence q can be approximated to 28.

ANALOGY: suppose you need to add 10...... trillion to 100 then obviously the result after addition can be approximated to 10......trillion.

Hence C
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Re: Which of the following best approximates the value of q if [#permalink]  01 Apr 2012, 23:25
Expert's post
Smita04 wrote:
Which of the following best approximates the value of q if 5^28 + 3^11 = 5^q ?

(A) 39
(B) 30
(C) 28
(D) 27
(E) 17

Similar questions to practice:
gmat-number-properties-exponents-84309.html
tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029229

Hope it helps.
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Re: Which of the following best approximates the value of q if [#permalink]  25 Jun 2013, 04:55
Expert's post
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Exponents: math-number-theory-88376.html

All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39
All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

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Re: Estimations [#permalink]  28 Dec 2013, 05:13
Bunuel wrote:

Which of the following best approximates the value of q if $$5^{28}+3^{11}=5^q$$?

A. 39
B. 30
C. 28
D. 27
E. 17

Note that we need approximate value of $$q$$. Now, $$5^{28}$$ is much, much, much bigger than $$3^{11}$$. So adding $$3^{11}$$ to $$5^{28}$$ will be closer to $$5^{28}$$ then to $$5^{29}$$, basically $$3^{11}$$ is negligible in this case. So $$q\approx{28}$$.

Bunuel, my approach for this question was:

I approximated $$3^{11}$$ to $$30^{10}$$ and thus $$3^{11}$$ = $$5*6^{10}$$ ----> factor ---> $$5^{10}$$ ($$5^{18}$$ + 6) .. Since the addition in the parenthesis is minescule, we can ignore it. And thus we have $$5^{10}$$ * $$5^{18}$$ = $$5^{28}$$ .... so q = 28

But obviously 30^10 is about a third of the size of 3^11 so I do not know if these operations work in general. Do they?
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Re: Estimations [#permalink]  28 Dec 2013, 05:17
Expert's post
aeglorre wrote:
Bunuel wrote:

Which of the following best approximates the value of q if $$5^{28}+3^{11}=5^q$$?

A. 39
B. 30
C. 28
D. 27
E. 17

Note that we need approximate value of $$q$$. Now, $$5^{28}$$ is much, much, much bigger than $$3^{11}$$. So adding $$3^{11}$$ to $$5^{28}$$ will be closer to $$5^{28}$$ then to $$5^{29}$$, basically $$3^{11}$$ is negligible in this case. So $$q\approx{28}$$.

Bunuel, my approach for this question was:

I approximated $$3^{11}$$ to $$30^{10}$$ and thus $$3^{11}$$ = $$5*6^{10}$$ ----> factor ---> $$5^{10}$$ ($$5^{18}$$ + 6) .. Since the addition in the parenthesis is minescule, we can ignore it. And thus we have $$5^{10}$$ * $$5^{18}$$ = $$5^{28}$$ .... so q = 28

But obviously 30^10 is about a third of the size of 3^11 so I do not know if these operations work in general. Do they?

It seems that you need to brush up fundamentals on this topic, because unfortunately very little above is correct.

Theory on Exponents: math-number-theory-88376.html

All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39
All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

Hope this helps.
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Re: Estimations [#permalink]  28 Dec 2013, 10:02
I solved it this way:

$$5^2^8+3^1^1=5^q$$

$$3^3=27$$ which is pretty close to $$5^2$$
$$3^1^1$$ is 3 times $$3^3$$ and one time $$3^2$$ thus we can approximate to 3 times $$5^2$$ and one 5

$$5^2^8+5^7=5^7(5^2^1+1)$$.

we can conclude that the best approximation is $$5^2^8$$ since $$5^2^1+1$$ is a little bit more than $$5^2^1$$. and $$5^7$$ times (a little bit more than $$5^2^1$$) = around $$5^2^8$$

hope it helps.
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Last edited by gmat6nplus1 on 29 Dec 2013, 03:18, edited 1 time in total.
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Which of the following best approximates the value of q if [#permalink]  29 Dec 2013, 02:34
Which of the following best approximates the value of q if $$5^{28} + 3^{11} = 5^q$$ ?
(A) 39
(B) 30
(C) 28
(D) 27
(E) 17

$$5^{28} + 3^{11} = 5^q$$

Or, $$5^{28}( 1 + \frac{3^{11}}{5^{28}}) = 5^q$$

Since, $$\frac{3^{11}}{5^{28}} << 1$$, the term $$(1 + \frac{3^{11}}{5^{28}})$$can be approximated to 1.

Thus, $$5 ^ {28} = 5 ^q$$

Or, $$q = 28$$.

Last edited by arunspanda on 29 Sep 2014, 04:48, edited 1 time in total.
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Re: Which of the following best approximates the value of q if [#permalink]  29 Dec 2013, 02:35
vwjetty wrote:
Which of the following best approximates the value of q if 5^28+3^11=5^q ?

A. 39
B. 30
C. 28
D. 27
E. 17

We have: 5^28+3^11=5^q
==> because 3^11 > 0 --> 5^q MUST be equal or greater than 5^28 ==> q MUST be equal or greater than 28
==> Option D and E are out immediately.

Divide both sides by 5^q and q >= 28
We have:
5^(28-q) + 3^11/5^q = 1

Because q >= 28 ==> 3^11/5^q = 0.0000xyz that is very small, we can ignore it.
Thus, 5^(28-q) must be approximate 1
==> 28-q = 0 ==> q is approximate 28

Hope it helps.
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Re: Estimations [#permalink]  11 Apr 2014, 20:29
Bunuel wrote:

Which of the following best approximates the value of q if $$5^{28}+3^{11}=5^q$$?

A. 39
B. 30
C. 28
D. 27
E. 17

Note that we need approximate value of $$q$$. Now, $$5^{28}$$ is much, much, much bigger than $$3^{11}$$. So adding $$3^{11}$$ to $$5^{28}$$ will be closer to $$5^{28}$$ then to $$5^{29}$$, basically $$3^{11}$$ is negligible in this case. So $$q\approx{28}$$.

I thought it this way:$$3^11=5^q-5^28$$.Now if we take$$q>28$$,we'll be able to take out $$5^28$$ common from RHS;that cannot equal $$3^11$$.And if we take $$q<28$$,We'll get -ve RHS;that again cannot equal $$3^11$$.
So the closest value can be $$q=28$$ only.But can this question be there in the GMAT?Even the closest/aprox. value in this case is quite far away!
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Re: Which of the following best approximates the value of q if [#permalink]  11 Apr 2014, 22:04
vwjetty wrote:
Which of the following best approximates the value of q if 5^28+3^11=5^q ?

A. 39
B. 30
C. 28
D. 27
E. 17

Raise to the power (1/10) to both the sides we will get:

5^2.8 + 3^1.1 = 5^q/10

3^1.1 is negligible as compared to 5^2.8 hence q = 28
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Re: Which of the following best approximates the value of q if [#permalink]  14 Apr 2014, 00:13
Option D & E can be neglected as the power would either be SAME or increase

Try to close in the powers of both the expressions

Consider $$5^{28}$$

$$= 5^{4*7}$$

$$= (5^4)^7$$

$$= 625^7$$

$$625^7$$ is a huge number as compared to $$3^{11}$$; so addition of the later will not have any significant effect to the exponent

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Re: Which of the following best approximates the value of q if [#permalink]  29 Sep 2014, 03:43
I solved it in this way:

5^28+3^11=5^q ?

3^2 = 27~ 5^2

So we can write 3^11 = 5^7

Now

5^28+5^7=5^q

5^7 (5^21 + 1) = 5^q => We can ignore 1 as it is much smaller than 5^21
5^28 = 5^q => q= 28 Hence C
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Re: Which of the following best approximates the value of q if [#permalink]  23 Feb 2015, 07:00
Karishma's explanation is great. Thanks Karishma. It helps to better visualize.
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Re: Which of the following best approximates the value of q if   [#permalink] 23 Feb 2015, 07:00
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