To simplify the given expression, we need to rationalize the denominators.
Let's start by simplifying the fractions:
9√33 = √(9^2 * 33) = √(99)
27√11 = √(27^2 * 11) = √(729 * 11) = √(7992)
Now, let's rewrite the expression with rationalized denominators:
(√99) / (√7992)
To rationalize the denominator, we need to find the prime factors of 7992.
The prime factorization of 7992 is: 2^3 * 3 * 7^2 * 11
Now, let's rewrite the expression with the rationalized denominator:
(√99) / (√(2^3 * 3 * 7^2 * 11))
To rationalize the denominator, we can multiply both the numerator and denominator by the square root of the missing prime factors.
(√99 * √(2^3 * 3 * 7^2 * 11)) / (√(2^3 * 3 * 7^2 * 11) * √(2^3 * 3 * 7^2 * 11))
Simplifying the denominator:
(√99 * √(2^3 * 3 * 7^2 * 11)) / (√(2^3 * 3 * 7^2 * 11 * 2^3 * 3 * 7^2 * 11))
Now, the square root of a product is equal to the product of the square roots of the factors.
(√99 * √(2^3 * 3 * 7^2 * 11)) / (√(2^3) * √(3) * √(7^2) * √(11) * √(2^3) * √(3) * √(7^2) * √(11))
Simplifying further:
(√99 * √(2^3 * 3 * 7^2 * 11)) / (2 * √(3) * 7 * √(11) * 2 * √(3) * 7 * √(11))
Now we can cancel out the common factors in the numerator and denominator:
(√99 * √(2^3 * 3 * 7^2 * 11)) / (2 * 7 * 2 * √(3) * √(11) * √(3) * √(11))
Simplifying further:
(√99 * √(2^3 * 3 * 7^2 * 11)) / (4 * 7 * √(3) * √(11) * √(3) * √(11))
Now, multiplying the remaining square roots:
(√99 * √(2^3 * 3 * 7^2 * 11)) / (4 * 7 * √(3 * 11 * 3 * 11))
Simplifying the numerator:
√(99 * 2^3 * 3 * 7^2 * 11) / (4 * 7 * √(33))
Now, simplifying the expression:
√(5544) / (4 * 7 * √(33))
Lastly, the square root of 5544 is equal to 74.4271887.
Plugging in the values:
74.4271887 / (4 * 7 * √(33))
Now we can multiply the numbers in the denominator:
74.4271887 / (28 * √(33))
Simplifying further, the answer is:
2.65 / √(33)
So, the simplified form of the expression is 2.65 / √(33).