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If 0 < a < 1, which of the following is the greatest?

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If 0 < a < 1, which of the following is the greatest?  [#permalink]

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New post 10 Sep 2017, 05:13
1
4
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A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

67% (01:31) correct 33% (01:27) wrong based on 141 sessions

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If 0 < a < 1, which of the following is the greatest?  [#permalink]

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New post 10 Sep 2017, 12:55
1
Bunuel wrote:
If 0 < a < 1, which of the following is the greatest?


A. \((-2a)^{(-2)}\)

B. \(\frac{1}{a^{(-2)}}\)

C. \((\frac{1}{a})^{(-2)}\)

D. \(a^{(-2)}\)

E. \(a^2\)


as \(a\) is positive and \(a<1\), so \(\frac{1}{a}>1\). Hence a fraction where \(a\) is in the denominator and numerator is \(1\) will have a value greater than \(1\). Hence we can straight away eliminate options B, C & E as \(a\) is in the numerator (essentially option B, C & E are same \(= a^2\) and as it is given \(a<1\), so squaring both sides will yield \(a^2<1\))

Option A: can be written as \(\frac{1}{(2a)^{2}}\) \(= \frac{1}{4a^2}\)

Option D: can be written as \(\frac{1}{a^2}\). Multiply the numerator and the denominator by \(4\). we get
\(\frac{4}{4a^2} > \frac{1}{4a^2}\)

Hence Option D
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If 0 < a < 1, which of the following is the greatest?  [#permalink]

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New post 14 Sep 2017, 11:09
Bunuel wrote:
If 0 < a < 1, which of the following is the greatest?

A. \((-2a)^{(-2)}\)

B. \(\frac{1}{a^{(-2)}}\)

C. \((\frac{1}{a})^{(-2)}\)

D. \(a^{(-2)}\)

E. \(a^2\)

Let a = \(\frac{1}{4}\), and evaluate each answer choice. Straight algebra -- reciprocals, fractions, and squares -- became too tangled.

A. \((-2a)^{(-2)}\) --> \(\frac{1}{(-2*a)^2}\)

\(\frac{1}{(-2*\frac{1}{4})^2}\) =

\(\frac{1}{(-\frac{1}{2})^2}\) = \(\frac{1}{(\frac{1}{4})}\) = 4


B. \(\frac{1}{a^{(-2)}}\) --> \(\frac{a^2}{1^2}\)

\(a^2\) = \((\frac{1}{4})^2\) =\(\frac{1}{16}\)


C.\((\frac{1}{a})^{(-2)}\) --> \(\frac{a^2}{1^2}\)=

\(a^2\). Same as Answer B =\(\frac{1}{16}\)


D. \(a^{(-2)}\)--> \(\frac{1}{a^2}\)

\(\frac{1}{(\frac{1}{4})^2}\) = \(\frac{1}{\frac{1}{16}}\) = 16


E. \(a^2\) = same as Answer B = \(\frac{1}{16}\)

ANSWER D
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Re: If 0 < a < 1, which of the following is the greatest?  [#permalink]

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New post 20 Sep 2017, 15:51
2
Bunuel wrote:
If 0 < a < 1, which of the following is the greatest?


A. \((-2a)^{(-2)}\)

B. \(\frac{1}{a^{(-2)}}\)

C. \((\frac{1}{a})^{(-2)}\)

D. \(a^{(-2)}\)

E. \(a^2\)


Since we know a is between 0 and 1, let’s let a = 1/2. Now we analyze each answer choice:

A)

[-2(1/2)]^-2 = (-1)^-2 = 1

B)

1/(1/2)^-2 = 1/(2^2) = ¼

C)

(1/(1/2))^-2 = 2^-2 = 1/4

D)

(1/2)^-2 = 2^2 = 4

E)

(1/2)^2 = 1/4

Answer: D
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Re: If 0 < a < 1, which of the following is the greatest?  [#permalink]

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New post 05 Dec 2019, 20:18
Bunuel wrote:
If 0 < a < 1, which of the following is the greatest?


A. \((-2a)^{(-2)}\)

B. \(\frac{1}{a^{(-2)}}\)

C. \((\frac{1}{a})^{(-2)}\)

D. \(a^{(-2)}\)

E. \(a^2\)


We see that both choices B and D can be simplified as a^2, which is choice E; thus, we know that none of these 3 choices could be the right answer. We are left with choices A and D.

We can let a = ½. Thus, choice A becomes (-1)^(-2) = 1 and choice D becomes (½)^(-2) = 2^2 = 4. We see that choice D is has the greatest value.

Answer: D
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Re: If 0 < a < 1, which of the following is the greatest?   [#permalink] 05 Dec 2019, 20:18
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