can someone help me solve this :?

Question

1/e^ -x^2 =(1/e^-3x)*e^ -2

Answer 1

Simplify each side first:

1/e^ -x^2
= e^(x^2)

(1/e^-3x)*e^ -2
= (e^3x) * e^-2
= e^(3x - 2)

So, now you have:
e^(x^2) = e^(3x - 2)
so...
x^2 = 3x - 2
x^2 - 3x + 2 = 0
(x - 2)(x - 1) = 0
x = 1 or 2

Answer 2

rewrite:
1 / [e^(-x^2)] = (e^-2) * 1 / [e^(-3x)]

a negative exponent indicates an inverse (one over something), such as:

2^(-2) is the same as 1/ (2^2)
Same thing if the negative exponent is in the denominator:;
1/ [2^(-2)] = 2^2

Rewrite, getting rid of the negative exponents:

e^(x^2) = e^(3x) / e^2

A division of powers (of the same base) is the same as subtracting exponents.
For example: (2^7)/(2^4) = 2^(7-4) = 2^3

e^(x^2) = e^(3x-2)

For this to be true, the exponents on each side must be the same:

x^2 = 3x - 2

x^2 -3x +2 = 0

(x-2)(x-1)=0
x = 2
and
x=1