To simplify the expression 4 sqrt 810 / 4 sqrt 2, we start by canceling out the common factor of 4 in the numerator and the denominator:
4 sqrt 810 / 4 sqrt 2 = sqrt 810 / sqrt 2
Next, we can use the property of square roots that states: sqrt(a) / sqrt(b) = sqrt(a / b). This allows us to combine the square roots:
sqrt(810 / 2)
Now, we simplify the division:
810 / 2 = 405
So, we have:
sqrt(405)
Next, we simplify sqrt(405). We can factor 405 into its prime factors:
405 = 5 x 81 = 5 x 9 x 9 = 5 x 9²
Next, we can write this as:
sqrt(405) = sqrt(5 x 9²) = sqrt(5) x sqrt(9²) = sqrt(5) x 9
This gives us:
9 sqrt(5)
Therefore, the simplest form of the quotient 4 sqrt 810 / 4 sqrt 2 is:
9 sqrt(5)