how would e^(x^(-2)) + [x(e^(x^(-2))(-2x^(-3)] simplify to be [e^(x^-2)]/x^2 ?
I tried simplifying it but i can't seem to get the answer...(wolframalpha gave the wrong answer...)
Repost with balanced parentheses
post it.
To simplify the given expression, let's break it down step by step:
Starting with the given expression:
e^(x^(-2)) + x(e^(x^(-2))(-2x^(-3))
Step 1: Rewrite the expression
Write the term x as x^1 to make the exponent notation consistent:
e^(x^(-2)) + x^1(e^(x^(-2))(-2x^(-3))
Step 2: Distribute the x^1
Multiply x^1 with each term inside the parentheses:
e^(x^(-2)) + (-2x^(-3))(x^1)(e^(x^(-2)))
Step 3: Simplify the exponents
Combine the exponents by adding them:
e^(x^(-2)) + (-2x^(-2-3))(e^(x^(-2)))
Simplify the exponent inside the parentheses:
x^(-2-3) = x^(-5)
So now the expression becomes:
e^(x^(-2)) + (-2x^(-5))(e^(x^(-2)))
Step 4: Combine like terms
Since both terms have the same base (e^(x^(-2))), you can add them together:
(1 - 2x^(-5))(e^(x^(-2)))
Step 5: Simplify further
To simplify the expression, let's rewrite it in a different form:
e^(x^(-2)) = 1 / e^(x^2) [taking reciprocal of base]
Rewriting the expression using the above substitution:
(1 - 2x^(-5))(1 / e^(x^2))
Simplifying the exponent inside the parentheses:
x^(-5) = 1 / x^5
Now the expression becomes:
(1 - 2(1 / x^5))(1 / e^(x^2))
Multiplying and simplifying further:
(1 - 2 / x^5) / e^(x^2)
Finally, simplifying the expression further:
(1 - 2 / x^5) e^(-x^2)
So, the simplified form of the given expression is:
[1 - 2 / x^5] e^(-x^2)
Note: The expression [e^(x^-2)] / x^2 doesn't appear to be equivalent to the original expression.