SOLUTION: Explain how you use the 5 rules of exponents to simplify the following expressions. Do not leave any negative exponents in your answer.: (3x^2 · y^2) · (-5x · y^4)^2

(3x^2 · y^2) · (-5x · y^4)^2 

(3x2y2)(-5xy4)2

The left parenthetical expression has no outer exponents so
we can erase the parentheses:

3x2y2(-5xy4)2

Write the remaining parenthetical expression as the bracketed
expression [(-5)xy4]2 since you need to group the negative sign
along with the 5 in parentheses for the -5 factor

3x2y2[(-5)xy4]2

Since the bracketed expression has an OUTER exponent, make sure
every factor inside the bracket has an INNER exponent showing. 
If it doesn't, give it an INNER exponent of 1.  We give (-5) and
x INNER exponents of 1 each.

3x2y2[(-5)1x1y4]2

Now remove the brackets by multiplying each INNER exponent by the
OUTER exponent 2.

3x2y2(-5)1·2x1·2y4·2

3x2y2(-5)2x2y8

Rearrange the factors:

3(-5)2x2x2y2y8

Write (-5)2 as (25)

3(25)x2x2y2y8

Multiply 3 by 25 getting 75

75x2x2y2y8

Add the exponents of the two x2's, getting x4

75x4y2y8

Add the exponents of the y2 and the y8, getting y10

75x4y10

Edwin
AnlytcPhil@aol.com