If \(\frac{4}{3^9} = a - \frac{2}{3^7} - \frac{2}{3^8} - \frac{2}{3^9}\), then what is the value of a?


A. 32/3^33

B. 10/3^8

C. 10/3^6

D. 10/3^2

E. 3
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If you don't see niks18 's shortcut, and if you keep your wits about you, this method takes much less than the average time:

\(\frac{4}{3^9} = a - \frac{2}{3^7} - \frac{2}{3^8} - \frac{2}{3^9}\)

**Multiply all terms by \(3^9\) =

\(4 = (3^9)a - (2)(3^2) - (2)(3^1) - 2\)

These numbers are not huge, thus:

\(4 = 3^9a - 18 - 6 - 2\)

\(4 + 26 = 3^9a\)

\(30 = 3^9a\)

\(\frac{3^1*10^1}{3^9}= a\)

\(\frac{10}{3^{(9-1)}}= \frac{10}{3^8}= a\)

Answer B

**
For the terms whose denominators are not "canceled" by \(3^9\):

\(\frac{a^{n}}{a^{m}} = a^{(n-m)}\)

\((3^9)*(\frac{2}{3^7})=\frac{2*3^9}{3^7}= (2)(3^{(9-7)})= (2)(3^2)\) AND

\((3^9)*(\frac{2}{3^8})=\frac{2*3^9}{3^8}= (2)(3^{(9-8)})= (2)(3^1)\)
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In the depths of winter, I finally learned
that within me there lay an invincible summer.
-- Albert Camus, "Return to Tipasa"

dave13 , I factored 30 because no answer choice has 30 as a numerator. Three choices have 10 as a numerator. I needed to get rid of the 3. (I wrote the exponents so that my next step would be clear.)

Second, I could have written \(3^{9+(-1)}\). Same thing.
When we have the same base (3), with different exponents (1 and 9), and we are dividing or multiplying, what do we do?(Hint: see my footnote)

Please take a look at Bunuel EXPONENTS

Purplemath,Exponents - Basic Rules, and Simplifying Expressions
Math Planet,Powers and Exponents

Hope that helps.
_________________

In the depths of winter, I finally learned
that within me there lay an invincible summer.
-- Albert Camus, "Return to Tipasa"