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26 Nov 2019, 01:36
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D
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Difficulty:
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Question Stats:
69% (02:26) correct
31%
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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
Are You Up For the Challenge: 700 Level Questions
A. 2
B. 3
C. 6
D. 12
E. 18
Are You Up For the Challenge: 700 Level Questions
26 Nov 2019, 04:28
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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
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
05 Dec 2019, 14:12
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LevanKhukhunashvili
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
\(\frac{(3^3)}{(2^9)}=3^3*2^{-9}\)
