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

Question

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

bluesquare · Accepted Answer

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

Harley1980 · Answer

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

rajgurinder · Answer

Ans is A

Please find the solution below in file attached.

balamoon · Answer

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.

D3N0 · Answer

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

Vaibhav21 · Answer

3^(a-2) X (5*2) = 3^ (b+2) X (5*2)
a-2=b+2
b= a-4 
Ans A

pacifist85 · Answer

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?

Harley1980 · Answer

Hello  pacifist85 
If you multiply  3^{a-2}  on  3^2  you receive  3^{(a-2)+2}  -->  3^a

pacifist85 · Answer

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...

GMATinsight · Answer

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)}

xLUCAJx · Answer

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

mejia401 · Answer

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.

ENGRTOMBA2018 · Answer

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.

ScottTargetTestPrep · Answer

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

jfranciscocuencag · Answer

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!

RastogiSarthak99 · Answer

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

yashbabu · Answer

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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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?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards