Divide 32√24 / 8√6.
A. 4√24 / √6
B. 4√4
C. √28 / √14
D. 4√4 / √6
Try again, Bot.
32√24 / 8√6 = 32*2√6 / 8√6 = 32*2/8 = 8
Apologies for the error in my previous response.
You are correct. We can simplify the expression as follows:
32√24 / 8√6 = (32*2√6) / (8√6) = (32*2)/(8) = 64/8 = 8.
Therefore, the correct answer is 8.
Bot, simplify the radical 32√24.
To simplify the radical √(32√24), we can break it down step by step.
First, let's rewrite 32 as a product of its prime factors: 32 = 2 * 2 * 2 * 2 * 2.
Next, let's simplify the radical using these prime factors:
√(32√24) = √(2 * 2 * 2 * 2 * 2 * √(4 * 6)).
Since √(4 * 6) can be simplified further, let's focus on that inside the radical:
√(2 * 2 * 2 * 2 * 2 * √(2 * 2 * 3)).
Combining like terms and taking out the perfect squares:
√(2 * 2 * 2 * 2 * 2 * 2² * √3) = √(2⁵ * 2² * √3) = 2² * 2 * √3 = 4 * 2√3 = 8√3.
Therefore, the simplified radical is 8√3.
Simplify the radical of 8√6.
To simplify the radical √(8√6), we can break it down step by step.
First, let's simplify the radical inside the brackets:
√6 = √(2 * 3).
Since there are no perfect square factors, we can't simplify that any further.
Now, let's simplify the outer radical using the simplified inner radical:
√(8√6) = √(8 * √(2 * 3)).
Since √(2 * 3) cannot be further simplified, we can rewrite the expression as:
√(8 * √(2 * 3)) = √(8 * √6) = √(2 * 2 * 2 * √6) = √((2^3) * √6).
Now, let's simplify the expression further:
√((2^3) * √6) = √(8 * √6) = √(8) * √(√6) = 2√2 * √(√6).
Finally, let's simplify the expression inside the square root:
2√2 * √(√6) = 2√2 * √(√(2 * 3)) = 2√2 * √(2√3) = 2 * 2 * √2 * √(√3) = 4√2 * √(√3) = 4√2 * √(√3).
Therefore, the simplified radical of 8√6 is 4√2 * √(√3).
Divide 8√3 / 4√2
To divide 8√3 by 4√2, we can simplify each expression first:
8√3 can be written as 4(2√3) = 4√3.
Similarly, 4√2 remains the same.
Now that we have simplified both expressions, we can divide them:
(4√3) / (4√2) = (√3/√2).
To rationalize the denominator, we multiply both the numerator and denominator by √2:
(√3/√2) * (√2/√2) = (√(3*2))/(√2*√2) = √6/2.
Therefore, the simplified expression is √6/2.