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# If 0.375^3 = 2^a3^b6^c, what is the value of b - a?

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Math Expert
Joined: 02 Sep 2009
Posts: 61302
If 0.375^3 = 2^a3^b6^c, what is the value of b - a?  [#permalink]

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26 Nov 2019, 00:36
00:00

Difficulty:

55% (hard)

Question Stats:

67% (02:40) correct 33% (02:33) wrong based on 39 sessions

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If $$0.375^3=2^a3^b6^c$$, what is the value of $$b - a$$?

A. 2
B. 3
C. 6
D. 12
E. 18

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Senior Manager
Joined: 13 Feb 2018
Posts: 498
GMAT 1: 640 Q48 V28
If 0.375^3 = 2^a3^b6^c, what is the value of b - a?  [#permalink]

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26 Nov 2019, 03:28
1
Let's try

1) lets convert the left-hand side into an integer form
$$0.375^3=(\frac{375}{1000})^3$$

2) prime factorize the result above
$$375=5^3*3$$
$$1000=2^3*5^3$$
Dont forget about the third power and we have the simplified left-hand side
$$\frac{(5^9*3^3)}{(2^9*5^9)}=\frac{(3^3)}{(2^9)}=3^3*2^{-9}$$

3) work on right-hand side
$$2^a*3^b*2^c*3^c$$
We can add the powers with same bases
$$2^{a+c}*3^{b+c}$$

So we have the equations
b+c=3
a+c=-9

b-a=12

IMO
ans: D
Intern
Joined: 09 Sep 2019
Posts: 38
Location: Switzerland
Concentration: Strategy, Economics
GPA: 3.1
Re: If 0.375^3 = 2^a3^b6^c, what is the value of b - a?  [#permalink]

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05 Dec 2019, 13:12
LevanKhukhunashvili wrote:
Let's try

1) lets convert the left-hand side into an integer form
$$0.375^3=(\frac{375}{1000})^3$$

2) prime factorize the result above
$$375=5^3*3$$
$$1000=2^3*5^3$$
Dont forget about the third power and we have the simplified left-hand side
$$\frac{(5^9*3^3)}{(2^9*5^9)}=\frac{(3^3)}{(2^9)}=3^3*2^{-9}$$

3) work on right-hand side
$$2^a*3^b*2^c*3^c$$
We can add the powers with same bases
$$2^{a+c}*3^{b+c}$$

So we have the equations
b+c=3
a+c=-9

b-a=12

IMO
ans: D

If one recognizes that
$$0.375^3=(\frac{3}{8})^3$$
time can be saved.
$$\frac{(3^3)}{(2^9)}=3^3*2^{-9}$$
Re: If 0.375^3 = 2^a3^b6^c, what is the value of b - a?   [#permalink] 05 Dec 2019, 13:12
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