To simplify the expression 9√33 ÷ 27√11, we can simplify both the numerator and denominator separately and then divide.
First, let's simplify the numerator, 9√33:
The prime factorization of 33 is 3 * 11.
So, we can rewrite 9√33 as 9√(3 * 11).
Since the square root of 3 is already simplified, we can rewrite 9√(3 * 11) as 9 * √3 * √11, which is equal to 9√(33).
Next, let's simplify the denominator, 27√11:
The prime factorization of 27 is 3 * 3 * 3.
So, we can rewrite 27√11 as 3 * 3 * 3 * √11.
Since the cube root of 3 is already simplified, we can rewrite 3 * 3 * 3 * √11 as 27 * √11.
Now we can rewrite the expression 9√33 ÷ 27√11 as (9√33) / (27√11) = (9√(33)) / (27√(11)) = (9/27) * (√(33) / √(11)).
Simplifying further:
(9/27) * (√(33) / √(11)) = (1/3) * (√(33) / √(11)).
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator: √(11) / √(11).
This gives us:
(1/3) * (√(33) / √(11)) * (√(11) / √(11)) = (1/3) * (√(33*11) / (√(11*11)) = (1/3) * (√(363) / 11).
So the simplified expression is (√363) / 33.
9√33 ÷ 27√11.
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