How do you simplify (8e^2/ 2e^5)^-1

To simplify `((8e^2)/(2e^5))^(-1)` , first simplify the fraction in parenthesis,

`(8e^2)/(2e^5) ` . We can simplify the numerical part by dividing both numerator and denominator by the common factor of 2:

`(8e^2)/(2e^5) = (4e^2)/(e^5) ` .

The rest can be simplified by using the rules of exponents: when dividing the powers of the same base, the exponents subtract. So,

`(4e^2)/(e^5) = 4e^(2 - 5) = 4e^(-3)` . This can also be written as `4/e^3` , using the definition of the negative exponent: `x^(-n) = 1/x^n` (A base raised to a negative power is the reciprocal of the same base raised to a positive power.)

Now, we also have to take the exponent -1 of this result. This can be done by again using the rules exponents:

`(4e^(-3))^(-1) = 4^(-1) e^3 = e^3/4` . Here, each factor inside the parenthesis is raised to the power of (-1).

So, the original expression simplifies as

`((8e^2)/(2e^5))^(-1) = e^3/4` .