Simplify the following quotient (sqrt(8m ^ 3 * n ^ 3))/(sqrt(2m ^ 2 * n ^ 2)) * (sqrt(6m ^ 2 * n ^ 2))/(sqrt(2m ^ 3 * n ^ 2)) point ) (2mn * sqrt(3n))/3; (6mn * sqrt(3n))/3; 2m * n ^ 3 * sqrt(3m); 48mn * sqrt(3n)

Answer 1

To simplify the given quotient, we need to multiply the numerators and denominators together using the properties of radicals. This will help us simplify and combine like terms.

First, let's simplify the numerator:

√(8m^3 * n^3) = √(8 * m * m^2 * n * n^2) = 2mn√(2m^2n^2)
√(6m^2 * n^2) = √(6 * m^2 * n^2) = √(2 * 3 * m^2 * n^2) = √(2m^2n^2)√3

So the numerator becomes:
2mn√(2m^2n^2) * √(2m^2n^2)√3 = 2mn * √(2m^2n^2) * √(2m^2n^2)√3 = 2mn * 2m^2n *√3 = 4m^3n^2√3

Now, let's simplify the denominator:

√(2m^2 * n^2) = √(2 * m^2 * n^2) = m√(2n^2) = mn√2

√(2m^3 * n^2) = √(2 * m^2 * m * n^2) = m√(2m * n^2) = mn√(2mn^2)

So the denominator becomes:
mn√2 * mn√(2mn^2) = m^2n^2√2 * √(2mn^2) = m^2n^2 * √2 * √(2mn^2) = 2m^2n^3

Now we can simplify the quotient by dividing the numerator by the denominator:

(4m^3n^2√3) / (2m^2n^3) = (4m^3n^2/2m^2n^3)√3 = 2m^(3-2)n^(2-3)√3 = 2mn^(-1)√3 = (2mn√3)/n

Therefore, (sqrt(8m^3 * n^3))/(sqrt(2m^2 * n^2)) * (sqrt(6m^2 * n^2))/(sqrt(2m^3 * n^2)) simplifies to (2mn * sqrt(3n))/n.

This can also be written as: (2m * n * sqrt(3n))/n = 2m * sqrt(3n).