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FACTORING AND EXPONENTS!!!

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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FACTORING AND EXPONENTS!!!

by CMP.1 » Fri Mar 23, 2012 5:22 am
I have been trying my utmost to learn the concepts behind these two questions, and ideally how to apply these so that any future questions can be handled with ease, yet I am still struggling.

I would be very very grateful if someone could please can someone show me, step by step, the procedures involved in solving these two questions, and explain the REASONING behind these steps?



(1) if (1/5)^m (1/4)^18 = 1/2(10)^35

m = ???

and (2) If 2^x - 2^x-2 = 3(2^13)

x = ???


Many thanks,


Chris
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by GyanOne » Fri Mar 23, 2012 6:07 am
Hi Chris

The way to do these kind of questions is to first figure out how the base numbers in both sides of the equations are related. In this case, 2 and 5 are a factor of 10. i.e 10 can be written as 2 x 5

Now the power to which 10 is raised is the same as the powers to which 2 and 5 are raised.

so if (1/10)^35 = (1/ (2 x 5))^35

Now, according to the question given,

(1/5)^m (1/4)^18 = 1/2(10)^35
= (1/5)^m (1/2)^36 = 1/2(10)^35
Now, equating the powers of the bases
= (1/5)^m (1/2)^36 = 1/2(2 x 5)^35
= (1/5)^m (1/2)^36 = 1/2 x (1/10)^35
= (1/5)^m (1/2)^36 = 1/2 x (1/2 x 5)^35
= (1/5)^m (1/2)^36 = 1/2 x (1/2)^35 x (1/5)^35
= 1/2 x (1/5)^m x (1/2)35 = 1/2 x (1/2)^35 x (1/5)^35
Which shows that m = 35

Now for teh second one
If 2^x - 2^(x-2) = 3(2^13) - (1)

let 2^x be y, then substituting in equation (1), we get

Also, as a^-2 = (1/a)^2, 2^(x-2) = 2^x X 2^-2 => y X (1/2)^2 => y/4

y - y/4 = 3(2^13)
3y/4 = 3(2^13)
y/4 = 2^13
y = 2^13 X 2^2
y = 2^15

y = 2^15 = 2^x
Thus x =15

Hope this helps
GyanOne
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by seal4913 » Sat Mar 24, 2012 4:37 pm
How did you get from 1/2(2 x 5)^35 to 1/2 x (1/10)^35
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by AMANBIR SINGH » Sun Mar 25, 2012 1:21 am
hi chris...
for the second question i can suggest u much simpler way..
2^x-2^(x-2)=3(2^13)
we take teh L.H.S
2^x-(2^x*2^-2)
taking common 2^x
2^x[(1-2^-2)]
2^x[(1-1/4)]
2^x[(3/4)]
2^x[3/(2^2)]
3[2^(x-2)]

comparing this with the R.H.S
we get,
x-2=13
that implies
x=15
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