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((2^4)^6)^8

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Carcass
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((2^4)^6)^8 [#permalink] New post   20 May 2020, 08:57
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Quantity A
Quantity B
\(((2^{4})^{6})^{8}\)
\(((8^6)^4)^2\)



A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
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Carcass
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Re: ((2^4)^6)^8 [#permalink] New post   20 May 2020, 08:57
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((2^4)^6)^8 [#permalink] New post   21 May 2020, 05:26
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Carcass wrote:
Quantity A
Quantity B
\(((2^{4})^{6})^{8}\)
\(((8^6)^4)^2\)



A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.


Key property: \((x^a)^b = x^{ab}\)
Extension of key property: \(((x^a)^b)^c = x^{abc}\)

Apply the properties to get:
QUANTITY A: \(((2^{4})^{6})^{8}=2^{192}\)
QUANTITY B: \(((8^6)^4)^2=8^{48}\)

In order to compare these values we need the same base.
Rewrite \(8\) as \(2^3\) to get:
QUANTITY A: \(2^{192}\)
QUANTITY B: \((2^3)^{48}\)

Simplify:
QUANTITY A: \(2^{192}\)
QUANTITY B: \(2^{144}\)

Answer: A

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Re: ((2^4)^6)^8 [#permalink] New post   04 Nov 2024, 15:13
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No calculator needed ver
A=\( ((2^6)^4)^8\)
B=\( (((2^3)^6)^4)^2\)
compare \(2^n\):
A=6*4*8
B=3 *6*8 OR (3*6*(4*2))
A>B

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Re: ((2^4)^6)^8 [#permalink]
04 Nov 2024, 15:13
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