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Difficulty:
95%
(hard)
Question Stats:
50% (02:00) correct
50%
(01:47)
wrong
based on 128
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Rank the following quantities in order, from smallest to biggest.
I. \(2^{120}\)
II. \(3^{72}\)
III. \(17^{30}\)
(A) I, II, III
(B) I, III, II
(C) II, I, III
(D) III, I, II
(E) III, II, I
I. \(2^{120}\)
II. \(3^{72}\)
III. \(17^{30}\)
(A) I, II, III
(B) I, III, II
(C) II, I, III
(D) III, I, II
(E) III, II, I
13 Jun 2024, 04:19
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Comparing I. and II.:
We have \(2^{120}\) and \(3^{72}\). Looking at the exponents, one notices that both 120 and 72 have 24 as a factor: 5*24=120 and 3*24=72
\((2^5)^{\frac{120}{5}} = 32^{24}\) and \((3^3)^{\frac{72}{3}} = 27^{24}\)
As the exponents are equal, we look at the bases to evaluate which is bigger: \(32>27\). Therefore \(2^{120}>3^{72}\)
Comparing I. and III.:
Reason for choosing to compare I. and III. next is because looking at the exponents of 120 and 30, respectively, both have 30 as a factor. 4*30=120 and 1*30=30.
\((2^4)^{120/4} = 16^{30}\) and \(17^{30}\)
Once again, as the exponents are the same, one looks at the bases to compare. \(17>16\), therefore \(17^{30} > 2^{120}\)
From smallest to biggest: II. , I. & III.
ANSWER C
We have \(2^{120}\) and \(3^{72}\). Looking at the exponents, one notices that both 120 and 72 have 24 as a factor: 5*24=120 and 3*24=72
\((2^5)^{\frac{120}{5}} = 32^{24}\) and \((3^3)^{\frac{72}{3}} = 27^{24}\)
As the exponents are equal, we look at the bases to evaluate which is bigger: \(32>27\). Therefore \(2^{120}>3^{72}\)
Comparing I. and III.:
Reason for choosing to compare I. and III. next is because looking at the exponents of 120 and 30, respectively, both have 30 as a factor. 4*30=120 and 1*30=30.
\((2^4)^{120/4} = 16^{30}\) and \(17^{30}\)
Once again, as the exponents are the same, one looks at the bases to compare. \(17>16\), therefore \(17^{30} > 2^{120}\)
From smallest to biggest: II. , I. & III.
ANSWER C
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