If (-5)^(4x) = 5^(9 + x) and x is an integer, what is the value of x ?

A. 5
B. 4
C. 3
D. 2
E. 1
_________________

since the values are equal
the powers can be equated.
we therefore have

4x=9-x
x=3

Since (-5)^4x = 5^(9+x)

Thus both sides are equal: Hence x>0

4x = 9+x
x = 3

Ans: 3

The important concept here is: (negative number)^(even integer) = some positive value
Examples
(-5)^2 = 25
(-3)^4 = 81
(-1)^10 = 1
(-2)^6 = 64

Also notice that:
(-5)^2 = (5)^2 = 25
(-3)^4 = (3)^4 = 81
(-1)^10 = (1)^10 = 1
(-2)^6 = (2)^6 = 64
The same can be said for ANY negative value raised to an EVEN power

Now onto the question.............
Since 4x is EVEN for any integer x, we can write:
(-5)^(4x) = (5)^(4x)

So, we can write: 5^(4x) = 5^(9 + x)
Since we now have the base, we can conclude that 4x = 9 + x
Solve to get: x = 3

Answer:

RELATED VIDEOS



_________________

simple and good question
First few seconds I tried solving as\(\frac{1}{5^{4x}}\)
Later, I substituted options
Any negative value raised to an Even power is always become Positive value
Thanks for explanations