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HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val

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Bunuel
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HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   24 Aug 2020, 20:00
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If \(3^b – (3^8 + 3^6) = 0\), what is the approximate value of b?

A. 5
B. 8
C. 10
D. 12
E. 14


 

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B
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Re: HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   24 Aug 2020, 20:11
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3^b - (3^8+3^6) = 0
3^b - 3^6(3^2+1) = 0
3^b - 3^6(3^2) = 0 {10 is nearest to 9 (3^2) as compared to 27 (3^3) }
3^b = 3^8

Answer is B

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Re: HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   24 Aug 2020, 20:14
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3^b - (3^8+3^6) = 0
3^b - 3^6(3^2+1) = 0
3^b = 3^6*10
b = 6+2 (approx)
b = 8

Hence (B)
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Re: HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   24 Aug 2020, 20:26
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3^b -(3^8+3^6) = 0 or, 3^b = 3^8 + 3^6

3^b = 3^6 (3^2 + 1)
3^b = 3^6 * 10...which is approximately 3^2

3^b = 3^ (6+2)
3^b = 3^8

B is the answer.

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Re: HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   24 Aug 2020, 21:35
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What is the approximate value of b?

\(3^b = (3^8 + 3^6)\)
\(3^b = (3^2 * 3^6 + 3^6)\)
\(3^b = 10 * 3^6\)

So, \(3^b\) must be between B and C. Let’s juxtapose:

\(3^b = 10 * 3^6\)
B. \(3^8 = 9 * 3^6\)
C. \(3^{10} = 81 * 3^6\)

We can see that \(3^b\) is closer to B than to C.

Hence B

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Re: HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   24 Aug 2020, 21:41
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3^b-(3^8+3^6)
3^b-3^6(9+1)
3^b-3^6*10

10~3*3 ---- 1
3^b-3^8 from 1
hence b=8
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Re: HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   24 Aug 2020, 21:45
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Given, 3^b–(3^8+3^6)=0
=> 3^b= (3^8+3^6)
=> 3^b= 3^6(3^2+1)
=> 3^b= 3^6(10)
=> 3^b= 3^6(9) ( As we need to approximate)
=> 3^b= (3^8)
=> b=8
Option B

We can also quickly verify:
=> 3^b= 3^6(10)
=> 3^b= 729*10
=> 3^b= 7290
Now, 3^8= 6561, and 3^9= 19683 (This is way too big than 7290)
Hence, 3^8 is the best approximate value.
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Re: HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   25 Aug 2020, 00:07
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Quote:
If 3^b–(3^8+3^6)=0, what is the approximate value of b?


Solution:
=> 3^b–3^6 (3^2+1)=0
=> 3^b–3^6 (10)=0
=> 3^6 (3^(b-6)–10)=0
divide both side by 3^6
=> 3^(b-6)–10 = 0
=> 3^(b-6) = 10
Consider 10 as 9. As we have been asked approximate value of B.
=> 3^(b-6) = 9
=> 3^(b-6) = 3^2
=> b-6 = 2
=> b = 8
whcih is option B.
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Re: HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   25 Aug 2020, 03:35
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If \(3^b–(3^8+3^6)=0\), what is the approximate value of b?

A. 5
B. 8
C. 10
D. 12
E. 14



\(3^b–(3^8+3^6)=0\) - expression cannot be simplified further.
questions asks "approximate" value for b.

\(3^8+3^6\) - we don't need to actually calculate this big numbers.
the value will be slightly higher than the 3^8
so b will be something like ~8.121232313
we won't sum up exponents here.

(B) is the answer. :heart
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Re: HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   02 Oct 2021, 18:40
The trick here is to approximate as it won't be feasible to try to calculate the exact value
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Re: HOT Competition: If 3^b – (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   02 Oct 2021, 19:33
TomNook wrote:
The trick here is to approximate as it won't be feasible to try to calculate the exact value

You only have to take the approximate value of 3^2+1 that is equal to 10
We know 3^3 = will be way higher than 10 and 3^1 is less than 10
So 3^2 is the only feasible option. Hence it won't take much time to solve question.

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Re: HOT Competition: If 3^b (3^8 + 3^6) = 0, what is the approximate val [#permalink] New post   16 Oct 2022, 09:08
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Re: HOT Competition: If 3^b (3^8 + 3^6) = 0, what is the approximate val [#permalink]
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