Simplify and State Restrictions: (5x^3y^2/27x) x (9xy/25x^2y^2)?

  5x³y²       9xy

---------- • ---------- =                     // Restrictions: x≠0, y≠0

    27x      25x²y²

// 5 goes into both 5 and 25. Cancel the 5 in the numerator

// with the 25 in the denominator leaving you 5/5 = 1 and 25/5 = 5

  1x³y²       9xy

---------- • ---------- =

    27x       5x²y²

// x² goes into both x³ and x². Cancel the x³ in the numerator

// with the x² in the denominator leaving you x³/x² = x and x²/x²=1

  1xy²       9xy

---------- • ---------- =

    27x       5(1)y²

// 9 goes into both 9 and 27. Cancel the 9 in the numerator

// with the 27 in the denominator leaving you 9/9 = 1 and 27/9 = 3

  1xy²       (1)xy

---------- • ---------- =

    3x        5(1)y²

// y² goes into y². Cancel y² in the numerator with y² in the

// denominator leaving you y²/y² = 1

  1x(1)       (1)xy

---------- • ---------- =

    3x        5(1)(1)

// x goes into x. Cancel x in the numerator with x in the

// denominator leavin you x/x = 1

  1(1)(1)       (1)xy

---------- • ---------- =

    3(1)        5(1)(1)

// There's nothing left to cancel, so multiply everything out

// in the numerator; multiply everything out in the denominator

  1(1)(1)(1)xy        xy

------------------ = --------  with restrictions x≠0, y≠0 .............ANS

    3(1)(5)(1)(1)     15

Presuming that is:

[5x³y² / (27x)] * [9xy / (25x²y²)]

I see that you multiplied the numerators and denominators to get:

5x³y²(9xy) / [(27x)(25x²y²)]

I would have simplified some things, first, but this is ok too.

From here, I would start canceling out common factors that show up in the numerator and denominator.  We have 5 from the numerator and in the 25 in the denominator and the 9 in the numerator and in the 27 in the denominator.  Canceling those out gives us:

x³y²(xy) / [(3x)(5x²y²)]

Now we can multiply the coefficients in the denominator and start to simplify the x's and y's multiplied by each other.  The product of two numbers with the same base is the same as the sum of the exponents. 

So the 3 and 1 from the x's in the numerator become a 4.  The 2 and 1 from the y's become a 3, etc.  After doing that, we get:

x³y²(xy) / (15x *x²y²)

x⁴y³ / (15x³y²)

finally, the quotient of two numbers with the same base is the same as subtracting the exponents.  So the x's become 1 and the y's become 1:

xy / 15

This is why I like to simplify before I multiply if possible.

Your mistake was to take 45/675 and reduce it to 9/75.

First, 9/75 isn't reduced, you can still divide both sides by 3 to get 3/25

But your simplify step was incorrect.  45/675 simplifies to 1/15.

So other than that, you would have gotten the correct answer.

And yes, your restrictions are that neither x nor y can be 0, otherwise you'll be dividing by 0 in the original expression, which you can't do.