9√33 ÷ 27√11.
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.
To simplify the expression 9√33 ÷ 27√11, we can start by simplifying each square root individually:
√33 can be simplified as follows:
- Write 33 as a product of its prime factors: 33 = 3 * 11.
- Take the square root of each prime factor: √33 = √(3 * 11).
- Simplify the square roots: √33 = √3 * √11.
√11 is already in its simplest form.
Now, let's substitute the simplified square roots back into the original expression:
9√33 ÷ 27√11 = 9(√3 * √11) ÷ 27√11.
Next, divide 9 by 27:
9(√3 * √11) ÷ 27√11 = (9/27)(√3 * √11) ÷ √11.
Simplify the fraction 9/27:
9/27 reduces to 1/3.
(1/3)(√3 * √11) ÷ √11 = (1/3)(√3 * (√11/√11)).
Multiply the square roots:
(1/3)(√3 * (√11/√11)) = (1/3)(√(3 * 11)/√11) = (1/3)(√33/√11).
Now, we can simplify √33/√11:
- Write √33/√11 as a single square root: √33/√11 = √(33/11).
- Simplify the square root of the fraction: √(33/11) = √(3).
Finally, substitute back into the expression:
(1/3)(√33/√11) = (1/3)(√3) = √3/3.
Therefore, 9√33 ÷ 27√11 simplifies to √3/3.
To simplify the expression 9√33 ÷ 27√11, we can start by simplifying the square roots separately.
First, let's simplify the numerator: 9√33.
To simplify the square root of 33, we need to determine if there are any perfect square factors. The factors of 33 are 1, 3, 11, and 33. None of these factors are perfect squares.
So, we can't simplify the square root of 33 any further. Therefore, the numerator remains the same: 9√33.
Now, let's simplify the denominator: 27√11.
To simplify the square root of 11, we again need to determine if there are any perfect square factors. The factors of 11 are 1 and 11. Since neither of these factors are perfect squares, we can't simplify the square root of 11 any further. So, the denominator remains the same: 27√11.
Now that we have simplified the numerator and the denominator, we can proceed to divide them.
Dividing 9√33 by 27√11 is the same as multiplying by the reciprocal of 27√11.
The reciprocal of 27√11 is 1/(27√11), which can be rewritten as √11/(27 * √11).
Now we can multiply:
(9√33) * (√11/(27 * √11)) = (9 * √33 * √11) / (27 * √11)
Since the square roots cancel out when multiplied in the numerator and the denominator, we are left with:
(9 * √(33 * 11)) / (27)
Simplifying further, we have:
(9 * √(363)) / (27)
The square root of 363 can be simplified by finding any perfect square factors. The factors of 363 are 1, 3, 11, 33, 121, and 363.
The perfect square factors are 1, 121 (11 squared).
So, we can simplify the square root of 363 as:
(9 * √(121 * 3)) / (27)
= (9 * √121 * √3) / 27
= (9 * 11 * √3) / 27
Now we can simplify the expression further:
= (99√3) / 27
Dividing both the numerator and the denominator by 9, we get:
= 11√3 / 3
So, 9√33 ÷ 27√11 simplifies to 11√3 / 3.