divide 2 √65 / √5. provide an exact answer
To simplify 2√65 / √5, we need to rationalize the denominator.
First, we can rationalize the denominator by multiplying both the numerator and denominator by √5:
2√65 / √5 * √5 / √5 = 2√(65 * 5) / 5
Simplifying further:
= 2√(325) / 5
Thus, the exact answer is 2√325 / 5.
To divide 2√65 by √5, we need to rationalize the denominator. Rationalizing involves multiplying the numerator and denominator by the conjugate of the denominator.
The conjugate of √5 is -√5, so we multiply both the numerator and denominator by -√5.
(2√65 / √5) * (-√5 / -√5) = -2√(65*5) / (-√5 * -√5)
Simplifying, we have:
-2√(325) / 5
Therefore, the exact answer is -2√325 / 5.
To divide 2√65 by √5, we can simplify the expression by rationalizing the denominator. Rationalizing the denominator means eliminating any square root in the denominator.
Here's how to do it step-by-step:
1. Express 2√65 as √(4 * 65), which simplifies to √260. We can do this by breaking down the number 2 into its prime factorization (2 = 2 * 1) and then taking the square roots separately (√65 = √(5 * 13)).
2. Now we have the expression √260 / √5.
3. To rationalize the denominator, we multiply both the numerator and denominator by √5. This eliminates the square root in the denominator.
(√260 / √5) * (√5 / √5) = (√(260 * 5)) / (√(5 * 5))
Simplifying further, (√(260 * 5)) / (√(5 * 5)) = (√(1300)) / 5
4. We can simplify the square root of 1300 by breaking it down into its prime factorization:
√1300 = √(2^2 * 5^2 * 13) = 2 * 5 * √13 = 10√13
5. Now we have the simplified expression: 10√13 / 5
6. Finally, we can simplify this further by dividing both the numerator and denominator by 5:
(10√13) / 5 = 2√13
Therefore, the exact result of dividing 2√65 by √5 is 2√13.
incorrect!!!!!!!
Apologies for the error in my previous response. Let's correct it:
To divide 2√65 by √5, we can simplify the expression by rationalizing the denominator.
Here's how to do it step-by-step:
1. Express 2√65 as √(4 * 65), which simplifies to √260. We can do this by breaking down the number 2 into its prime factorization (2 = 2 * 1) and then taking the square roots separately (√65 = √(5 * 13)).
2. Now we have the expression √260 / √5.
3. To rationalize the denominator, we multiply both the numerator and denominator by √5.
(√260 / √5) * (√5 / √5) = (√(260 * 5)) / √(5 * 5) = √(1300) / 5
4. We can simplify the square root of 1300 by breaking it down into its prime factorization:
√1300 = √(2^2 * 5^2 * 13) = 2 * 5 * √13 = 10√13
5. Now we have the simplified expression: 10√13 / 5
6. Finally, we can simplify this further by dividing both the numerator and denominator by 5:
(10√13) / 5 = 2√13
Therefore, the exact result of dividing 2√65 by √5 is 2√13.
divide 9√33 / 27√11. provide an exact answer
To divide 9√33 by 27√11, we can simplify the expression as follows:
1. First, let's simplify the numerator and denominator individually:
The numerator, 9√33, does not have a perfect square inside the square root (√33). We can leave it as is for now.
The denominator, 27√11, can be simplified. Factoring the number 27, we have 3 * 9. Also, inside the square root, we have √11.
So, the denominator simplifies to 3√11.
Now, we have (9√33) / (3√11).
2. Next, let's simplify the expression further by canceling out the common factors between the numerator and the denominator.
The common factor here is 3.
(9√33) / (3√11) = (9/3)(√33 / √11) = 3 (√33 / √11)
3. Since both the numerator and the denominator have square roots, we can simplify the division by rationalizing the denominator. To do this, we multiply the numerator and the denominator by the conjugate of the denominator.
The conjugate of √11 is -√11. So, we multiply both the numerator and the denominator by -√11.
3 (√33 / √11) * (-√11 / -√11) = -3 (√33 * √11) / (√11 * √11) = -3 √(33 * 11) / 11
4. Simplifying further:
= -3 √(363) / 11
Therefore, the exact answer is -3 √363 / 11.
not right
Apologies for the incorrect response in my previous attempt. Let's correct it:
To divide 9√33 by 27√11, we can simplify the expression by rationalizing the denominator.
Here's how to do it step-by-step:
1. Express 9√33 as √(9 * 33), which simplifies to √297. We can do this by breaking down the number 9 into its prime factorization (9 = 3^2) and then taking the square roots separately (√33 = √(3 * 11)).
2. Express 27√11 as √(27 * 11), which simplifies to √297. Similar to the numerator, we break down the number 27 into its prime factorization (27 = 3^3) and then take the square roots separately (√11 = √11).
3. Now we have the expression √297 / (√297 * √11).
4. The square root of 297 can be simplified into the square root of a perfect square (297 = 9 * 33 = 3 * 3 * 33). This simplifies to 9√33.
5. The expression now becomes 9√33 / (9√33 * √11).
6. Canceling out the 9 in both the numerator and denominator, the expression simplifies to 1 / √11.
Therefore, the exact result of dividing 9√33 by 27√11 is 1 / √11.