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If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?

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If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  08 Jun 2015, 08:58
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If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?

(A) a – 4
(B) a – 2
(C) a + 4
(D) 3a + 2
(E) 3a + 4
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  10 Jun 2015, 02:15
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If \(3^a + 3^{(a - 2)} = (90)(3^b)\), what is b in terms of a?

\(3^a + 3^{(a - 2)} = (3^2*10)(3^b)\) -->
\(3^a(1 + 3^{(-2)}) = (3^{(2+b)})(10)\) -->
\(3^a\frac{10}{9} = (3^{(2+b)})(10)\)-->
\(3^a = (3^{(2+b)})(9)\)-->
\(3^a = (3^{(2+b)})(3^2)\)-->
\(3^a = (3^{(4+b)})\)-->
\(a=4+b\)-->
\(b=a-4\) or A
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  08 Jun 2015, 09:20
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Bunuel wrote:
If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?

(A) a – 4
(B) a – 2
(C) a + 4
(D) 3a + 2
(E) 3a + 4


Kudos for a correct solution.


\(3^a + 3^{a - 2} =3^{a-2}(3^2+1)=3^{a-2}*10\)

\((90)(3^b)=3^2*10*3^b=3^{b+2}*10\)

so \(3^{a-2}*10=3^{b+2}*10\)

\(b+2=a-2\) --> \(b = a-4\)

Answer is A
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  08 Jun 2015, 09:58
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Ans is A

Please find the solution below in file attached.
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  08 Jun 2015, 11:00
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If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?


Lets take a=2. 3^2+3^0=(90)3^b -> 10/90=3^b --> 3^-2=3^b ---> b=-2.

Substitute a=2 in answer choices, only A fits correctly.

Thanks,

Please give me kudos.
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  09 Jun 2015, 01:48
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Bunuel wrote:
If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?

(A) a – 4
(B) a – 2
(C) a + 4
(D) 3a + 2
(E) 3a + 4


Ans : A

Solution: generalizing both sides gives us

3^a-2 * 5*2 = 3^ b+2 * 5*2
means a-2=b+2
b= a-4
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  14 Jun 2015, 04:42
3^(a-2) X (5*2) = 3^ (b+2) X (5*2)
a-2=b+2
b= a-4
Ans A
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  17 Jun 2015, 04:18
Harley1980 wrote:
Bunuel wrote:
If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?

(A) a – 4
(B) a – 2
(C) a + 4
(D) 3a + 2
(E) 3a + 4


Kudos for a correct solution.


\(3^a + 3^{a - 2} =3^{a-2}(3^2+1)=3^{a-2}*10\)

\((90)(3^b)=3^2*10*3^b=3^{b+2}*10\)

so \(3^{a-2}*10=3^{b+2}*10\)

\(b+2=a-2\) --> \(b = a-4\)

Answer is A


Hello could you explain the part in red to me plz?
I understand that you took 3^a-2 as a common factor. But how did you get that taking the power of "a-2" from a power of "a" leaves a square?
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  17 Jun 2015, 04:28
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pacifist85 wrote:
Harley1980 wrote:
Bunuel wrote:
If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?

(A) a – 4
(B) a – 2
(C) a + 4
(D) 3a + 2
(E) 3a + 4


Kudos for a correct solution.


\(3^a + 3^{a - 2} =3^{a-2}(3^2+1)=3^{a-2}*10\)

\((90)(3^b)=3^2*10*3^b=3^{b+2}*10\)

so \(3^{a-2}*10=3^{b+2}*10\)

\(b+2=a-2\) --> \(b = a-4\)

Answer is A


Hello could you explain the part in red to me plz?
I understand that you took 3^a-2 as a common factor. But how did you get that taking the power of "a-2" from a power of "a" leaves a square?


Hello pacifist85
If you multiply \(3^{a-2}\) on \(3^2\) you receive \(3^{(a-2)+2}\) --> \(3^a\)
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  17 Jun 2015, 04:55
Oh ok I think i see it now. So, you basally want to take 3^a-2 out, as a common factor. You have 3^a, and from that power (a) you are subtracting a-2, as you have already taken this as a fator.

Then you would have 3^a-(a-2) = 3^a-a+2 = 3^2. I hope I got it right...
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  17 Jun 2015, 05:04
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pacifist85 wrote:
Harley1980 wrote:
Bunuel wrote:
If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?

(A) a – 4
(B) a – 2
(C) a + 4
(D) 3a + 2
(E) 3a + 4


Kudos for a correct solution.


\(3^a + 3^{a - 2} =3^{a-2}(3^2+1)=3^{a-2}*10\)

\((90)(3^b)=3^2*10*3^b=3^{b+2}*10\)

so \(3^{a-2}*10=3^{b+2}*10\)

\(b+2=a-2\) --> \(b = a-4\)

Answer is A


Hello could you explain the part in red to me plz?
I understand that you took 3^a-2 as a common factor. But how did you get that taking the power of "a-2" from a power of "a" leaves a square?


Hi pacifist,

Just for your information and better understanding here are Rules of exponents

The powers of the numbers add up for the same base when the numbers are multiplied i.e. \(a^x * a^y = a^{(x+y)}\)
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  01 Jul 2015, 15:27
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Harley1980 wrote:
Bunuel wrote:
If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?

(A) a – 4
(B) a – 2
(C) a + 4
(D) 3a + 2
(E) 3a + 4


Kudos for a correct solution.


\(3^a + 3^{a - 2} =3^{a-2}(3^2+1)=3^{a-2}*10\)

\((90)(3^b)=3^2*10*3^b=3^{b+2}*10\)

so \(3^{a-2}*10=3^{b+2}*10\)

\(b+2=a-2\) --> \(b = a-4\)

Answer is A


I am confused on the first step you took that factored out the \((3^2+1)\)
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  01 Jul 2015, 17:29
xLUCAJx wrote:
Harley1980 wrote:
Bunuel wrote:
If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?

(A) a – 4
(B) a – 2
(C) a + 4
(D) 3a + 2
(E) 3a + 4


Kudos for a correct solution.


\(3^a + 3^{a - 2} =3^{a-2}(3^2+1)=3^{a-2}*10\)

\((90)(3^b)=3^2*10*3^b=3^{b+2}*10\)

so \(3^{a-2}*10=3^{b+2}*10\)

\(b+2=a-2\) --> \(b = a-4\)

Answer is A


I am confused on the first step you took that factored out the \((3^2+1)\)


Hi xLUCAJx,

They factored out the whole expression, \(3^{a-2}\), so that the \((3^{2}+1)\), or \(10\), would be isolated. Thinking forward, not one but two steps ahead, and it saved them from a fraction.
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  01 Jul 2015, 17:44
xLUCAJx wrote:
Harley1980 wrote:
Bunuel wrote:
If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?

(A) a – 4
(B) a – 2
(C) a + 4
(D) 3a + 2
(E) 3a + 4


Kudos for a correct solution.


\(3^a + 3^{a - 2} =3^{a-2}(3^2+1)=3^{a-2}*10\)

\((90)(3^b)=3^2*10*3^b=3^{b+2}*10\)

so \(3^{a-2}*10=3^{b+2}*10\)

\(b+2=a-2\) --> \(b = a-4\)

Answer is A


I am confused on the first step you took that factored out the \((3^2+1)\)


Think of it in this way:

\(3^a+ 3^{a - 2} =3^{a+2-2}+3^{a-2}=3^{a-2}*3^{2}+ 3^{a-2} = 3^{a-2}(3^2+1)=3^{a-2}*10\)

Going from 1st part to the second, we added and subtracted 2 from a as we need the power of a-2 common. For the second part, we used the property \(a^{x+y} = a^x*a^y\), where x= a-2 and y = 2.

I hope this clears your doubt.
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  21 Jun 2018, 16:58
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Bunuel wrote:
If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?

(A) a – 4
(B) a – 2
(C) a + 4
(D) 3a + 2
(E) 3a + 4


Simplifying, we have:

3^a + 3^a x 3^-2 = (90)(3^b)

3^a(1 + 3^-2) = 90(3^b)

3^a(1 + 1/9) = 90(3^b)

3^a(10/9) = 90(3^b)

3^a = 90(3^b) x 9/10

3^a = 9 x 3^b x 9

3^a = 81 x 3^b

3^a = 3^4 x 3^b

3^a = 3^(4 + b)

a = 4 + b

a - 4 = b

Answer: A
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  11 Feb 2019, 20:11
Good night!

Could someone please explain to me the following?


\(3^a + 3^{(a - 2)} = (3^2*10)(3^b)\)

\(3^a(1 + 3^{(-2)}) = (3^{(2+b)})(10)\)

From here...

Can we always just equal the powers of 3?

a - 2 = 2 + b

a - 4 = b???

Kind regards!
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  11 Aug 2022, 06:45
While others have clearly done better ways of doing this, I would just like to point out that in exponents, it is easy to miss how to take out common items.

So always take common smaller units. Like with 3^a + 3^a-2 --> write as 3^a-2( 3^(a-(a-2) + 1) --> 3^a-2 * 10
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  15 Aug 2022, 11:48
Take a= 4
3^4 + 3^(4-2) = 3^(2+b) x 10

3^2 (3^2+1) = 3^(2+b) x 10

3^2 (10) = 3^(2+b) x 10
3^2 = 3^(2+b)

2=2+b
b=0


put a=4 in options only A goes with b=0
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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a? [#permalink] New post  02 Sep 2023, 00:47
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