1. Simplify the expression ((5x^-2)/(4x^2))^-2

2. (x)/(x^2-81)-(9)/(81-x^2) =

3. Let (-3,3) and (-6,_2) be points in the Cartesian plane. Find the distance between the points.

((5x^-2)/(4x^2))^-2

= (5x^-2)(4x^2)^2
= (5x^-2)(16x^4)
= 80x^2

2. Do you realize that 81 - x^2 = -(x^2 - 81)

so (x)/(x^2-81)-(9)/(81-x^2)
= x/(x^2 - 81) + 9/(x^2 - 81)
= (x-9)/((x+9)(x-9))
= 1/(x+9) , x ≠ 9

for #3

you should have a formula for the distance between 2 points.

1. To simplify the given expression ((5x^-2)/(4x^2))^-2, we can follow these steps:

a. Start by applying the negative exponent to each term separately. This means moving both the numerator and denominator to the opposite side of the fraction and changing the sign of the exponent. So, ((5x^-2)/(4x^2))^-2 becomes ((4x^2)/(5x^2)) ^2.
b. After moving the terms, we can eliminate common factors in the numerator and denominator. Here, the common factor is x^2. Canceling out the x^2, we have ((4)/(5))^2.
c. Finally, evaluate the expression by simply squaring the fraction ((4)/(5))^2. This results in (16/25).

2. In this problem, we need to simplify the given expression:

(x)/(x^2-81)-(9)/(81-x^2)

To simplify it, we can follow these steps:
a. Start by noticing that the denominators are both difference of squares. Rewrite the expression with the difference of squares formula.
(x)/((x-9)(x+9))-(9)/(-(x-9)(x+9))

b. Now, we need to find a common denominator to add the two fractions. Multiply the first fraction by (-1)/(-1) to make the denominators the same.
(x)/((x-9)(x+9))-(9)/((x-9)(x+9))

c. Combine the fractions over the common denominator.
[x-9-9]/((x-9)(x+9))

d. Simplify the numerator.
[x-18]/((x-9)(x+9))

Therefore, the simplified expression is (x-18)/((x-9)(x+9)).

3. To find the distance between two points in the Cartesian plane, we can use the distance formula. The formula for the distance between two points (x1, y1) and (x2, y2) is given by:

Distance = √[(x2 - x1)^2 + (y2 - y1)^2]

Let's calculate the distance between (-3,3) and (-6,-2) using the distance formula:

Distance = √[(-6 - (-3))^2 + (-2 - 3)^2]
= √[(-6 + 3)^2 + (-2 - 3)^2]
= √[(-3)^2 + (-5)^2]
= √[9 + 25]
= √34

So, the distance between the points (-3,3) and (-6,-2) is √34, which is an irrational number.