Simplify the following quotient. Please show all the steps to find the answer.
√8m^5n^6 / √2m^3n^2 • √6m^4n^4 / √2m^3n^2
To simplify the quotient of the given terms, we can combine the like terms in the numerator and denominator and cancel them out if possible.
√8m^5n^6 / √2m^3n^2 • √6m^4n^4 / √2m^3n^2
First, let's simplify each individual square root:
√8m^5n^6 = (√(2^3) * m^(5/2) * n^(6/2)
= (√2^3) * m^(5/2) * n^3
= 2 * m^(5/2) * n^3
√2m^3n^2 = (√2) * m^(3/2) * n
√6m^4n^4 = (√6) * m^2 * n^2
Now, let's substitute these simplified square roots back into the original equation:
(2 * m^(5/2) * n^3) / (√2 * m^(3/2) * n) • (√6 * m^2 * n^2) / (√2 * m^(3/2) * n^2)
Next, we can cancel out any similar terms in the numerator and denominator:
(2 * m^(5/2) * n^3 * m^2 * n^2) / (m^(3/2) * n * m^(3/2) * n^2)
Merging the similar terms, we get:
(2 * m^(9/2) * n^5) / (m^(3/2) * n^(3/2))
Now, let's simplify the exponents:
(2 * m^((9/2) - (3/2))) * n^((5 - 3)/2)
= (2 * m^6 * n^(2/2))
= (2 * m^6 * n)
Therefore, the simplified quotient is 2m^6n.