Find the product of 5–√

Question

and 1/6
. What type of number is it?(1 point)
Responses

2.4027 . . . ; an irrational number
2.4027 . . . ; an irrational number

2.40; a rational number
2.40; a rational number

0.3726 . . . ; an irrational number
0.3726 . . . ; an irrational number

0.37; a rational number
0.37; a rational number

Answer 1

To find the product of \( 5 - \sqrt{5} \) and \( \frac{1}{6} \), we first calculate \( 5 - \sqrt{5} \).

Step 1: Calculate the value of \( 5 - \sqrt{5} \)

Using the approximate value of \( \sqrt{5} \): \[ \sqrt{5} \approx 2.236 \] Thus: \[ 5 - \sqrt{5} \approx 5 - 2.236 = 2.764 \]

Step 2: Multiply by \( \frac{1}{6} \)

Now, let's find the product: \[ (5 - \sqrt{5}) \cdot \frac{1}{6} \approx 2.764 \cdot \frac{1}{6} \approx 0.46067 \]

Step 3: Determine the exact value

However, to find the exact value: \[ \left(5 - \sqrt{5}\right) \cdot \frac{1}{6} = \frac{5 - \sqrt{5}}{6} \]

Step 4: Classify the number

Since \( \sqrt{5} \) is an irrational number, \( 5 - \sqrt{5} \) is also an irrational number. The product of an irrational number and a rational number (in this case, \( \frac{1}{6} \)) is also an irrational number.

Conclusion

Thus, the final product is \( \frac{5 - \sqrt{5}}{6} \), which is an irrational number.

Among the response options given, the closest numerical approximation is \( 0.37; \) and it is classified as "an irrational number."

So the answer is: 0.37; an irrational number.