CAT 1 - No Solution N

[jeremy.hansen]

Even when reading the explanations I'm confused about how to solve this problem and I am confident I would not be able to recognize or solve in another form. Hoping for prehaps a more basic understanding of what the problem is asking, how to recognize, how to attack and perhaps additional study direction.



Each of the following equations has at least one solution EXCEPT

A) -2^n = (-2)^-n

B) 2^-n = (-2)^n

C) 2^n = (-2)^-n

D) (-2)^n = -2^n

E) (-2)^-n = -2^-n

[adiagr]

Even when reading the explanations I'm confused about how to solve this problem and I am confident I would not be able to recognize or solve in another form. Hoping for prehaps a more basic understanding of what the problem is asking, how to recognize, how to attack and perhaps additional study direction.



Each of the following equations has at least one solution EXCEPT

A) -2^n = (-2)^-n

B) 2^-n = (-2)^n

C) 2^n = (-2)^-n

D) (-2)^n = -2^n

E) (-2)^-n = -2^-n

Hi,

Fisrt of all just check (D) and (E), is there any typo?

Now coming to the approach.

This is a bit confusing, but patiently you can solve this question. Basically this question is checking following fundas:

1. Properties of Indices. Specifically

(i) x^(-n) = {1/(x^n)} and

(ii) (x^a) / (x^b) = x^(a-b)

2. A negative No. raised to even Powers will give a Positive No. whereas a negative No. raised to odd powers will give a Negative No.

(-2) x (-2)x(-2) x (-2) = (-2)^4 = 16 (a Positive qty)

(-2) x (-2)x(-2) = (-2)^3 = -8 (a negative qty)

3. x^0 = 1;




I will illustrate by solving example (1)

(-2)^n = (-2)^-n ............given

From above rearranging the Right hand side

(-2)^-n = [1/{(-2)^n}]

Cross-multiplying

=> {(-2)^n} x {(-2)^n} = 1

= {(-2)^2n} =1

Right hand side can be written as (-2)^0

n=0; thus solution exists.

Similarly work out for other options also.

[jeremy.hansen]

nt

[mschwrtz]

Hey Jeremy,

You might find the properties of exponents easier to take on board if you see them written out normally. Look to page 68 of your Number Properties guide for more familiar versions of the rules to which adiagr refers.

Where s/he writes, x^(-n) = {1/(x^n)}, see the sixth rule in the table.

Where s/he writes, (x^a) / (x^b) = x^(a-b), see the third rule in the table.

The best bet for problems like this one is to know the relevant properties of exponents. Most people don't have very strong intuitions about the meanings of those exponents that can't be simply translated into multiplication problems, so you need enough recent practice with these expressions to solve them more or less mechanically, should the problem demand that.

If you can't do that, you might be able to use POE even still. Did the question stipulate that n>0? If not, try n=0, and see which answers that eliminates (any answer that is true for any of n is wrong). There must be some such constraint, though, or perhaps a typo, because n=0 makes all of A-C true, and n=any odd integer makes D and E true.