1. Sqrt 14r * 2 sqrt 4r

Is this 4r sqrt 14?

2. (sqrt 64 y^10 h^5)/(sqrt 16 y^12 h^3)

Is this (sqrt 4h^2)/(y^2)

#1 is correct

#2 is correct, but you left out the last step:

2h/y

I don't understand what you mean by the last step. Is the answer supposed to be 2h/y?

I get it--the square root.

To simplify these expressions, we can apply the rules of simplifying square roots.

1. Sqrt(14r) * 2 sqrt(4r)

To simplify, we can combine the square roots and multiply the coefficients:

sqrt(14r) * 2 sqrt(4r) = 2 sqrt(14r) sqrt(4r)

Since the square root of 4 is 2, we can simplify further:

2 sqrt(14r) sqrt(4r) = 2 * 2 sqrt(14r) sqrt(r)

Using the property sqrt(a) * sqrt(b) = sqrt(a * b), we can simplify further:

2 * 2 sqrt(14r) sqrt(r) = 4 sqrt(14r * r)

Now, multiplying r and r gives us r^2:

4 sqrt(14r * r) = 4 sqrt(14r^2)

Thus, the simplified expression is:

4 sqrt(14r^2)

2. (sqrt(64 y^10 h^5)) / (sqrt(16 y^12 h^3))

To simplify this expression, we can again combine the square roots and divide them:

(sqrt(64 y^10 h^5)) / (sqrt(16 y^12 h^3)) = sqrt(64 y^10 h^5 / (16 y^12 h^3))

Using the property sqrt(a/b) = sqrt(a) / sqrt(b), we can simplify further:

sqrt(64 y^10 h^5 / (16 y^12 h^3)) = sqrt(64 h^5 / (16 h^3)) * sqrt(y^10 / y^12)

Since 64/16 = 4 and h^5 / h^3 = h^2, we can simplify further:

sqrt(64 h^5 / (16 h^3)) * sqrt(y^10 / y^12) = sqrt(4h^2) * sqrt(y^10 / y^12)

Finally, simplifying the square roots gives us:

sqrt(4h^2) * sqrt(y^10 / y^12) = 2h * sqrt(y^10 / y^12)

Since y^10 / y^12 = 1 / y^2, we can simplify further:

2h * sqrt(y^10 / y^12) = 2h * sqrt(1 / y^2)

Using the property sqrt(1/a) = 1 / sqrt(a), we get:

2h * sqrt(1 / y^2) = 2h * (1 / y)

Thus, the simplified expression is:

(2h) / y