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Algebra Strategies question

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by gmattesttaker2 » Sun Feb 03, 2013 12:05 pm
Hello,

I was just wondering if you can please assist here. This question is from MGMAT Strategy Guide2 (P. 125)

(((3/4)^3))^(1/4) > 3/4

True or False

Answer: True


My approach was as follows:

3/4 approximates to 4/4

So, ((1)^3)^(1/4) = 1^(3/4) = 1^1 = 1

So,1 > 1 is False

Can you please assist here? Thanks a lot.


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Sri
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by Brent@GMATPrepNow » Sun Feb 03, 2013 12:31 pm
gmattesttaker2 wrote:Hello,

I was just wondering if you can please assist here. This question is from MGMAT Strategy Guide2 (P. 125)

(((3/4)^3))^(1/4) > 3/4

True or False

Answer: True
Phew, that's a lot of nested brackets! In fact, there are a few extras here.
Perhaps adding some square brackets will help. We get:
Is [(3/4)^3]^(1/4) > 3/4?

To begin, we can apply the power of a power rule to get:
Is (3/4)^(3/4) > 3/4?

If recognize that 0 < 3/4 < 1, we can see where (3/4)^(3/4) fits in the following pattern:
(3/4)^4 = 81/256
(3/4)^3 = 27/64
(3/4)^2 = 9/16
(3/4)^1 = 3/4
(3/4)^(3/4) = some value between 3/4 and 1
(3/4)^0 = 1
(3/4)^(-1) = 4/3
(3/4)^(-2) = 16/9

This means that (3/4)^(3/4) is greater than 3/4.

Cheers,
Brent
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by Brent@GMATPrepNow » Sun Feb 03, 2013 12:34 pm
There's also a "rule" that says,
"If 0<x<1 and y<1, then x^y > x"

So, (3/4)^(3/4) > 3/4

Cheers,
Brent
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by GMATGuruNY » Sun Feb 03, 2013 6:02 pm
gmattesttaker2 wrote:Hello,

I was just wondering if you can please assist here. This question is from MGMAT Strategy Guide2 (P. 125)

(((3/4)^3))^(1/4) > 3/4 ?

True or False
Another approach:

[(3/4)³]¹´� > (3/4) ?

To clear the fractional exponent, raise each side to the fourth power:
(3/4)³ > (3/4)�

Distribute the exponents:
3³/4³ > 3�/4�

Cross-multiply:
3³4� > 3�4³

Divide each side by 3³4³:
4 > 3.

Thus, it is TRUE that the lefthand side is greater than the righthand side.
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