Question

Find the product of 12−−√ and 56. What type of number is it?(1 point)
Pick one of the following Responses

4.1569 . . . ; an irrational number
4.1569 . . . ; an irrational number

4.2974 . . . ; an irrational number
4.2974 . . . ; an irrational number

2.8867 . . . ; an irrational number
2.8867 . . . ; an irrational number

2.6307 . . . ; an irrational number

Answer 1

To find the product of \(\sqrt{12}\) and \(56\), we can first calculate \(\sqrt{12}\):

\[
\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}
\]

Now, we can compute the product of \(2\sqrt{3}\) and \(56\):

\[
2\sqrt{3} \times 56 = 112\sqrt{3}
\]

Next, we will find the approximate value of \(112\sqrt{3}\). The approximate value of \(\sqrt{3}\) is about \(1.732\).

Now let's calculate:

\[
112 \times 1.732 \approx 193.984
\]

Now we need to check the type of this number. Since \(\sqrt{3}\) is irrational, \(112\sqrt{3}\) is also irrational because the product of a rational number (112) and an irrational number (\(\sqrt{3}\)) is irrational.

Thus, the final result is:

- The product \(112\sqrt{3}\) is approximately \(193.984\) which is not listed among the given responses because they seem to be close approximations of numbers much smaller than \(193.984\) and also irrational.

In conclusion, since none of the given options match our calculated approximation, the original question might be incorrect or misrepresented compared to the options provided. Based on the given responses, a check for a valid product calculation might suggest a reinterpretation of what is being asked.

Therefore, the straightforward answer we derived is that \(112\sqrt{3}\) is indeed **an irrational number**, regardless of the approximate numerical representations provided.