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

Question

Responses

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

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

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

2.8867 . . . ; an irrational number

Answer 1

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

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

Now, we multiply this by \( 56 \):

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

Next, let's approximate \( \sqrt{3} \):

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

Thus,

\[ 112\sqrt{3} \approx 112 \cdot 1.732 \approx 193.184 \]

Since \( 112\sqrt{3} \) involves \( \sqrt{3} \), which is an irrational number, the product \( 112\sqrt{3} \) is also irrational.

Now, by looking at the options provided:

1. \( 4.1569... ; an irrational number \)
2. \( 2.6307... ; an irrational number \)
3. \( 4.2974... ; an irrational number \)
4. \( 2.8867... ; an irrational number \)

None of these values seem to match \( 112\sqrt{3} \approx 193.184 \).

Nevertheless, we conclude that \( 112\sqrt{3} \) is an irrational number, but it doesn't match any of the numerical responses given. Thus, the product of \( \sqrt{12} \) and \( 56 \) is an irrational number.