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

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

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New post 26 Nov 2019, 01:36
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A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

63% (02:51) correct 37% (02:30) wrong based on 27 sessions

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

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New post 26 Nov 2019, 04:28
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
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Re: If 0.375^3 = 2^a3^b6^c, what is the value of b - a?  [#permalink]

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New post 05 Dec 2019, 14:12
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


Thanks for the helpful solution!

If one recognizes that
\(0.375^3=(\frac{3}{8})^3\)
time can be saved.
As it directly leads to
\(\frac{(3^3)}{(2^9)}=3^3*2^{-9}\)
GMAT Club Bot
Re: If 0.375^3 = 2^a3^b6^c, what is the value of b - a?   [#permalink] 05 Dec 2019, 14:12
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If 0.375^3 = 2^a3^b6^c, what is the value of b - a?

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