Expert: Scott A Wilson - 12/11/2010

Question
1)divide and then simplify by taking roots, if possible. assume all expressions under radicals represent positive numbers. (sqrt)2025xy divided by 3(sqrt)3
2)multiply and simplify by factoring. assume that all expressions under radicals represent non-negative numbers. (sqrt)12a^15b *(sqrt)8a^16b^14

Answer
1) Given this, √(2025xy)/(3√3), it is known that 2025 is 45², so this is 45√(xy)/(3√3).  Since 45 = 3*3*5, and 3=√3√3, 45 can be said to be 45 = (3√3)*5√3.

From this, we have 45√(xy)/(3√3) = ((3√3)*5√3)√(xy)/(3√3).

We can divide both the numerator and the denominator by 3√3 and get 5√3√(xy).  Combining the squareroots gives 5√(3xy).

2) To multiply, lets do the constant first.

It can be seen that 12*8 = 96, and 96 = 16*6, so √96 = √16√6 = 4√6.

For the variable a, it can be seen that a^15*a^16 = a^31 = a^30*a,
so the squareroot of this is a^15√a.

For the variable b, it is known to be √b√b^14 = (√b)b^7.
This can be rewritten as b^7√b.

Combine the answers at the end of the last three paragraphs to get the answer.  Note that outside the squareroot is a 4, a^15, and b^7,
all of these multiplied together.  Inside the squareroot is
6, a, and b, so it is √(6ab).