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Math Expert V
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If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?  [#permalink]

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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

Kudos for a correct solution.

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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?  [#permalink]

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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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GMAT 1: 660 Q48 V33 GMAT 2: 740 Q50 V40 If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?  [#permalink]

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1
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 $$3^{a-2}*10=3^{b+2}*10$$

$$b+2=a-2$$ --> $$b = a-4$$

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

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Ans is A

Please find the solution below in file attached.
Attachments Capture2.PNG [ 53.61 KiB | Viewed 3867 times ]

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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?  [#permalink]

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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]

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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]

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

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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 $$3^{a-2}*10=3^{b+2}*10$$

$$b+2=a-2$$ --> $$b = a-4$$

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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GMAT 1: 660 Q48 V33 GMAT 2: 740 Q50 V40 Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?  [#permalink]

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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 $$3^{a-2}*10=3^{b+2}*10$$

$$b+2=a-2$$ --> $$b = a-4$$

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]

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

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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 $$3^{a-2}*10=3^{b+2}*10$$

$$b+2=a-2$$ --> $$b = a-4$$

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. $$a^x * a^y = a^{(x+y)}$$
Attachments

File comment: www.GMATinsight.com Rules of Exponents.jpg [ 197.99 KiB | Viewed 3606 times ]

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Re: If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?  [#permalink]

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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 $$3^{a-2}*10=3^{b+2}*10$$

$$b+2=a-2$$ --> $$b = a-4$$

I am confused on the first step you took that factored out the $$(3^2+1)$$
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If 3^a + 3^(a - 2) = (90)(3^b), what is b in terms of a?  [#permalink]

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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 $$3^{a-2}*10=3^{b+2}*10$$

$$b+2=a-2$$ --> $$b = a-4$$

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]

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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 $$3^{a-2}*10=3^{b+2}*10$$

$$b+2=a-2$$ --> $$b = a-4$$

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]

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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

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

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