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Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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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

Originally posted by vwjetty on 23 Aug 2010, 08:32.
Last edited by Bunuel on 13 Dec 2017, 00:02, edited 3 times in total.
Edited the question.
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Re: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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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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Math Expert V
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Re: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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13

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: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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1
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 5^28+3^11=5  [#permalink]

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Math Expert V
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Re: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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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: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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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: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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2
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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Originally posted by gmat6nplus1 on 28 Dec 2013, 10:02.
Last edited by gmat6nplus1 on 29 Dec 2013, 03:18, edited 1 time in total.
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Re: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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2
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, $$q = 28$$.

Originally posted by arunspanda on 29 Dec 2013, 02:34.
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 5^28+3^11=5  [#permalink]

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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: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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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 5^28+3^11=5  [#permalink]

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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 5^28+3^11=5  [#permalink]

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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 5^28+3^11=5  [#permalink]

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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 5^28+3^11=5  [#permalink]

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VeritasPrepKarishma wrote:
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.

great explanation...I tried to solve it by logic..and eliminated right away D and E. A is way too much. Between B and C, went with B..now I see why I did the mistake..
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Re: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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mvictor wrote:
VeritasPrepKarishma wrote:
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.

great explanation...I tried to solve it by logic..and eliminated right away D and E. A is way too much. Between B and C, went with B..now I see why I did the mistake..

B could've been easily eliminated as for increasing another power of 5, you will need to atleast multiply that number by 5 atleast. 3^11 can be approximated to (3^6)^2 and 3^6 will give you 5^4 so you will end up getting almost 5^8 and this when added to 5^28 will still not do anything.
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GRE 1: Q169 V154 Re: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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Here is how i solved it up=>
5^q=5^28+3^11
I didn't realise that 3^11 was negligible when saw this statement.
So i followed as -->
3^3=5^2(approx)
hence 5^q=(5)^28+5^7(approx)=> 5^75^21+1
Here i neglected 1 and wrote 5^q=5^28
hence q=28(approx)

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Re: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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40 seconds approach:

Let's see if there is a pattern:

5^1 + 3^1 = 8 (Roughly close to 5^1)
5^2 + 3^2 = 34 (Roughly close to 5^2)
5^3 + 3^3 = 152 (Roughly close to 5^3)
...
...
and so on. We can see that 3 is pretty much negligible in this case and doesn't really add much to the power of 5. Therefore:
5^28 + 3^11 is close enough to 5^28 or option C!
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Re: Which of the following best approximates the value of q if 5^28+3^11=5  [#permalink]

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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

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

We can PLUG IN THE ANSWERS, which represent the approximate value of q.

B: q=30
$$5^{30} - 5^{28} = 3^{11}$$
$$5^{28}(5^2 - 1) = 3^{11}$$
$$5^{28}(24) = 3^{11}$$
Not viable.
The left side is MUCH GREATER than the right side.
Eliminate B.

D: q=27
$$5^{27} - 5^{28} = 3^{11}$$
Not viable.
The left side is negative and thus is MUCH SMALLER than the right side.
Eliminate D.

Since B makes the left side too big, while D makes the left side too small, the correct answer must be BETWEEN B AND D.

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Available for tutoring in NYC and long-distance. Re: Which of the following best approximates the value of q if 5^28+3^11=5   [#permalink] 16 Apr 2020, 16:31

# Which of the following best approximates the value of q if 5^28+3^11=5  