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

Question

Responses

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
2.6307 . . . ; an irrational number

4.1569 . . . ; an irrational number

Answer 1

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

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

Now we can calculate the product:

\[ \sqrt{12} \cdot 56 = 2\sqrt{3} \cdot 56 = 112\sqrt{3} \]

Next, we can find the numerical approximation of \(112\sqrt{3}\):

\[ \sqrt{3} \approx 1.732 \]

Now, we calculate:

\[ 112 \cdot 1.732 \approx 193.184 \]

Thus, \(112\sqrt{3} \approx 193.184\). This number is clearly irrational, since the product of a rational number (112) and an irrational number (\(\sqrt{3}\)) is always irrational.

Since none of the given numerical responses match this value of \(112\sqrt{3}\), it's clear that the question did not contain the correct answer in its options. However, we confirm that the value \(112\sqrt{3}\) is indeed an irrational number.

Thus, the product of \(\sqrt{12}\) and \(56\) is approximately \(193.184\) and it is an irrational number.