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.
Rewrite the irrational cube root 3^√189 as the product of an integer and another irrational cube root. Show your work.
To rewrite the irrational cube root as the product of an integer and another irrational cube root, we need to find a perfect cube factor of 189.
By examining the prime factorization of 189, we can see that it can be expressed as 3 * 3 * 3 * 7.
Therefore, we can rewrite 189 as (3^3) * 7.
Now, let's rewrite the original expression using this factorization:
3^(√189) = 3^(√((3^3) * 7))
Using the property of radicals, we can split the radical:
= 3^((√(3^3)) * (√7))
Simplifying further,
= 3^(3 * (√7))
Now, we have our desired form:
3^(√189) = 3^(3 * (√7)).
Therefore, the irrational cube root 3^√189 can be rewritten as the product of an integer (3^3 = 27) and another irrational cube root (√7), giving us 27√7.
First, let's simplify the expressions:
32√24 can be simplified as 4√(2*4*6) = 4(2√6) = 8√6.
Similarly, 8√6 can be simplified as 4(√6).
Now that we have simplified both expressions, we can divide them:
(8√6) / (4(√6)) = (8/4)(√6/√6) = 2(1) = 2.
Therefore, the answer is not one of the given options.
Apologies for the mistake.
To divide 32√24 / 8√6, we can simplify each term first:
32√24 can be simplified as 4(8√6) = 32√6.
8√6 can be simplified as 4(2√6) = 8√6.
Now we can divide:
32√6 / 8√6 = 32/8 * √6/√6 = 4 * 1 = 4.
Therefore, the correct answer is D. 4√4 / √6.