I am really having problems with monomials: Please help!

Simplify:

4x^2 y^3 Z
----------- (divided by)
5y^4 Z^3

is the answer
4x
------ (divided by)
5yz^2

With monomials, you can look at each unknown (variable) individually when you do a division:

4x²
------- = 4x²/5
5

y³/y4 = 1/y

z/z³ = 1/z²

So putting everything together:
4x^2 y^3 Z / 5y^4 Z^3
4x²/(5yz²)

Explain the difference in answers between the following exponential expressions. Include the solutions?

7^2, -7^2,(-7)^2, and 7^-2

To simplify the given expression (4x^2y^3Z) / (5y^4Z^3), you can follow these steps:

Step 1: Divide the coefficients:
Divide the coefficient in the numerator (4) by the coefficient in the denominator (5). The result is (4 / 5).

Step 2: Divide the variable with exponents:
Divide the variable with exponents in the numerator (x^2y^3Z) by the variable with exponents in the denominator (y^4Z^3).

- For the variable 'x', there is no corresponding variable 'x' in the denominator, so you simply leave it as it is.
- For the variable 'y', subtract the exponent in the denominator from the exponent in the numerator: y^3 / y^4 = y^(3-4) = y^(-1). Since y^(-1) is the same as 1/y, the result is 1/y.
- For the variable 'Z', subtract the exponent in the denominator from the exponent in the numerator: Z / Z^3 = Z^(1-3) = Z^(-2). Since Z^(-2) is the same as 1/Z^2, the result is 1/Z^2.

So, after dividing the variables with exponents, the result is (x^2 * 1 / y * 1 / Z^2).

Combining both the coefficient and the variables, the simplified expression is (4x / 5yz^2).