A very Smart Question Rather :

To solve or Compare the two Quantities we will try to get the Fraction ( In particular the Denominator Same for both Qty )

For Qty A we have the Following Situation :

99^ 9 / 9 ^ 99 = ( 11 * 9 ) ^ 9 / 9 ^99 --> 9^9(11^9)/ 9 ^ 99 --> 11^9 / 9 ^ 90 [ We divided or Took 9^ 9 out as Common ] .

So Both Qty are equal

ANSWER C

Key Concept: \((b^k)(c^k)=(bc)^k\)

We can also write: \((bc)^k=(b^k)(c^k)\)

Notice that we can rewrite 99 as (9)(11)
So....

GIVEN:
QUANTITY A: \(\frac{99^9}{9^{99}}\)

QUANTITY B: \(\frac{11^9}{9^{90}}\)


Rewrite 99 as (9)(11) to get:
QUANTITY A: \(\frac{[(9)(11)]^9}{9^{99}}=\frac{(9^9)(11^9)}{9^{99}}=\frac{11^9}{9^{90}}\)

QUANTITY B: \(\frac{11^9}{9^{90}}\)


Answer: C

Cheers,
Brent
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The trick is to identify that 99 = 11 * 9. This will enable us to rewrite Quantity A using 11 and 9 , making it possible to make meaningful comparisons.

Of course, you should know that \((x*y)^n = x^n * y^n\) and \(x^{n+m} = x^n * x^m\).


Quantity A = \(\frac{99^9}{9^{99}}\) = \(\frac{(11 * 9)^9}{9^{99}}\) = \(\frac{11^9 * 9^9}{9^{99 }}\) = \(\frac{11^9 * 9^9}{9^{90} * 9^9}\) = \(\frac{11^9}{9^{90}}\)

Quantity B = \(\frac{11^9}{9^{90}}\)

Both quantities are equal. The answer is C.
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