To find the product of \( \sqrt{12} \) and 13, we first calculate \( \sqrt{12} \).
We can simplify \( \sqrt{12} \) as follows:
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
Now, we can multiply this by 13:
\[ \text{Product} = 13 \times \sqrt{12} = 13 \times 2\sqrt{3} = 26\sqrt{3} \]
Next, we need to find the numerical value of \( 26\sqrt{3} \). We'll approximate \( \sqrt{3} \):
\[ \sqrt{3} \approx 1.732 \]
Thus:
\[ 26\sqrt{3} \approx 26 \times 1.732 = 44.992 \]
Since this is an approximate value, we will look also at the type of number. The product \( 26\sqrt{3} \), which involves \( \sqrt{3} \) (an irrational number), is itself irrational.
Now, let's match the options given:
- \( 4.1569...; \text{ an irrational number} \)
- \( 2.6307...; \text{ an irrational number} \)
- \( 4.2974...; \text{ an irrational number} \)
- \( 2.8867...; \text{ an irrational number} \)
None of the options we computed corresponds to our product \( 26\sqrt{3} \). The correct reasoning is acknowledged that the product is an irrational number, but, unfortunately, none of the given options match the actual value of \( 26\sqrt{3} \).
However, the important outcomes from the above calculations are:
- The product \( 26\sqrt{3} \) is indeed an irrational number.
- Its approximate value is around 44.992.
The accurate conclusion is that it is not listed among the given options, but we can affirm that \( 26\sqrt{3} \) is irrational nonetheless.