If n is a positive integer, then ((-3)^n)^-4 + (3^-n)^4 + ((-3)^-n)^4

If n is a positive integer, then \(((-3)^n)^-4 + (3^-n)^4 + ((-3)^-n)^4 = \)

Now all are raised to even power 4, so we can take base as 3.
If n is a positive integer, then \(((-3)^n)^{-4} + (3^{-n})^4 + ((-3)^{-n})^4 = \)If n is a positive integer, then\( ((3)^n)^{-4} + (3^{-n})^4 + ((3)^{-n})^4 = 3*3^{-4n}=3^{-4n+1}\)

C
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n = positive integer
((-3)^n)^-4 + (3^-n)4 + ((-3)^-n)^4 =
(-3)^-4n + 3^-4n + (-3)^-4n ------------------------(A)
As (-3)^-4n = (-1/3)^4n
(-1/3)^4n + (1/3)^4n +(-1/3)^4n -------------------------------(B)
As n is a positive integer, 4n will always be even
So, (-1/3)^4n = (1/3)^4n
Applying this result in expression (B),
(1/3)^4n + (1/3)^4n + (1/3)^4n =
3 x (1/3)^4n
3 x 1/3^4n
3^1-4n, So 3^-4n + 1
Imo. C

ANSWER:C
((-3)^n)^-4 + (3^-n)^4 + ((-3)^-n)^4
=(-3)^-4n+(3)^-4n+(-3)^-4n
=1/(-3)^4n+1/(3)^4n+1/(-3)^4n
=3/(3)^4n=(3)^1-4n

The easiest way is the test any positive integer for n say 2.
The equation then becomes \(3/9^4= 1/3^7\)
Test the same n=2 on the choices. Only choice C matches.

Answer: C

GIVEN If n is a positive integer, then ((-3)^n)^-4 + (3^-n)^4 + ((-3)^-n)^4 =
which can re written as
(1/(-3)^n ) ^4 + (1/3^n)^4 + (1/(-3)^n) ^4
at n =1
(1/(-3))^4 + (1/3)^4 + (1/(-3)) ^4
1/81+1/81+1/81 ; 3/81 ; 1/27 ( target value )
check with given options ,
for option C
3^(-4n+ 1) at n= 1
3^-3 ; 1/3^3 ; 1/27 same as our target value

OPTION C


If n is a positive integer, then ((-3)^n)^-4 + (3^-n)^4 + ((-3)^-n)^4 =
A (1/3)^(-4n) ;
B (1/3)^(-4n+1) ;
C 3^(-4n+ 1)
D 3^(-4n)
E 3^(-4n -1)

If n is a positive integer, then ((-3)^n)^-4 + (3^-n)^4 + ((-3)^-n)^4 =

A (1/3)^(-4n)
B (1/3)^(-4n+1)
C 3^(-4n+ 1)
D 3^(-4n)
E 3^(-4n -1)


Solution:

\((-3)^n)^-4 + (3^-n)^4 + ((-3)^-n)^4 \)

\(= ((-3)^-4n + (3^-4n) + ((-3)^-4n\)

\(= ((-1)^-4n *(3)^-4n + (3^-4n) + ((-1)^-4n * (3)^-4n) \)

\(= (3)^-4n + (3^-4n) + (3)^-4n) \)

\(= 3*(3^-4n) \)

\(= (3^-4n+1)\)

So, option C is correct

Let N = 1
=>> {(-3)^n)^-4 + (3^-n)^4 + (-3)^-n)^4}
=>> {(-3)^1)^-4 + (3^-1)^4 + (-3)^-1)^4}
=>> {(-3)^-4 + (1/3)^4 + (1/-3)^4}
=>> {(1/-3)^4 + (1/3)^4 + (1/-3)^4}
=>> {1/81+1/81+1/81}
=>> {3/81} =>>> 1/27 = L.H.S

Putting value of N = 1 in options:

A (1/3)^(-4n)
=> 1/3^-4
=> 3^4 => 81 (≠1/27)

B (1/3)^(-4n+1)
=> 1/3^(-4+1)
=> 1/3^(-3)
=> 3^3 => 27 (≠1/27)

C 3^(-4n+ 1)
=> 3^(-4+1)
=> 3^-3
=> 1/3^3 => 1/27 (=1/27)

D 3^(-4n)
=> 3^-4
=> 1/3^4
=> 1/81 (≠1/27)

E 3^(-4n -1)
=> 3^(-4-1)
=> 3^-5
=> 1/3^5
=> 1/243 (≠1/27)

Answer is C

Posted from my mobile device
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Asked: If n is a positive integer, then ((-3)^n)^-4 + (3^-n)^4 + ((-3)^-n)^4 =

A (1/3)^(-4n)
B (1/3)^(-4n+1)
C 3^(-4n+ 1)
D 3^(-4n)
E 3^(-4n -1)

\(((-3)^n)^-4 + (3^-n)^4 + ((-3)^-n)^4 = (-1)^{-4n}(3)^{-4n} + 3^{-4n} + (-1)^{-4n}(3)^{-4n} = 3^{-4n} + 3^{-4n} + 3^{-4n} = 3*3^{-4n} = 3^{1-4n}\);
Since \((-1)^{-4n} = 1\)

IMO C

C , imo

((-3)^n)^-4 + (3^-n)^4 + ((-3)^-n)^4 = (1/ 3^4n) + (1/ 3^4n) + (1/ 3^4n) = 3 / 3^4n = 3 ^ (1-4n)

Hence c .

Any number raised to a even power will result in a positive value.
Thus, ignore the negative sign of 3 in the equation. ----(1)

First Part: \(((-3)^{n})^{-4} = (-3)^{-4n} = \frac{1}{(-3)^{4n}} = \frac{1}{3^{4n}}...\)(implement (1))

Second Part: \((3^{-n})^{4} = (3)^{-4n} = \frac{1}{3^{4n}}\)

Third Part: \(((-3)^{-n})^{4} =(-3)^{-4n} = \frac{1}{(-3)^{4n}} = \frac{1}{3^{4n}}...\)(implement (1))

Adding all three parts together = \(\frac{1}{3^{4n}}+\frac{1}{3^{4n}}+\frac{1}{3^{4n}}\)
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad = \frac{3}{3^{4n}}\)
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad = \frac{1}{3^{4n-1}}\)
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad = 3^{-(4n-1)}\)

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad = 3^{-4n+1}\)

The significance of mentioning "n is a positive integer" in the question is:

Any number raised to a even power will result in a positive value.
Thus, we can ignore the negative sign of 3 in the equation

Had that not been mentioned, then n could be 3/4.... n=3/4
and \(4n = 4*3/4 = 3\)

Thus (-3) will have an odd power and you will not be able to neglect the negative sign.

Forgot to mention this in my previous post.

IMO ans is C

As all three simplified to \(3^(^-^4^n^)\);
Thus \(3^(^-^4^n^)\) x 3
\(3^(^-^4^n^+^1^)\)

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