Find the product of 12−−√ and 56. What type of number is it?(1 point) Responses 2.8867 . . . ; an irrational number 2.8867 . . . ; an irrational number 2.6307 . . . ; an irrational number 2.6307 . . . ; an irrational number 4.2974 . . . ; an irrational number 4.2974 . . . ; an irrational number 4.1569 . . . ; an irrational number

1 answer

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

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

Now we compute the product: \[ \sqrt{12} \times 56 = 2\sqrt{3} \times 56 = 112\sqrt{3} \]

Next, we need to determine the type of number \( 112\sqrt{3} \). Since \( \sqrt{3} \) is an irrational number, and any non-zero rational number multiplied by an irrational number results in an irrational number, \( 112\sqrt{3} \) is also an irrational number.

Now, for a numerical approximation of \( 112\sqrt{3} \): \[ \sqrt{3} \approx 1.732 \] Thus, \[ 112\sqrt{3} \approx 112 \times 1.732 \approx 193.184 \]

The product \( 112\sqrt{3} \) is irrational.

None of the provided responses match the calculated product, but we can affirm that:

  • The product \( 112\sqrt{3} \) is an irrational number.

Therefore, the conclusion is:

  • The product is an irrational number. (Knowing the numerical values did not lead to a match from given options.)