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

Question

Responses

2.6307 . . . ; an irrational number
2.6307 . . . ; an irrational number

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

Answer 1

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

We can simplify \( \sqrt{12} \): \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]

Next, we multiply this by \( 56 \): \[ 56 \times \sqrt{12} = 56 \times 2\sqrt{3} = 112\sqrt{3} \]

Now, we can calculate \( 112\sqrt{3} \) approximately, knowing that \( \sqrt{3} \approx 1.732 \): \[ 112 \times 1.732 \approx 193.184 \] The product \( 112\sqrt{3} \) is not a rational number because \( \sqrt{3} \) is an irrational number. Therefore, the product is also irrational.

None of the options you've provided with specific numerical values match this result, as they seemed to be different approximations. However, the conclusion is still valid:

The product \( 112\sqrt{3} \) is an irrational number. If you need a more specific numerical approximation from either the responses or further calculations, you might want to check the calculations again, but the type of number is indeed irrational.