What type of number will result from multiplying −8–√

Question

and 4/5
?(1 point)
Responses

an irrational number
an irrational number

the number zero
the number zero

a rational number
a rational number

an integer

Answer 1

To determine the type of number that results from multiplying \(-8 - \sqrt{8}\) and \(\frac{4}{5}\), we first need to simplify the expression.

1. Calculate the first term \(-8 - \sqrt{8}\): \[ -8 - \sqrt{8} = -8 - 2\sqrt{2} \quad (\text{since } \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}) \]

2. Multiply by \(\frac{4}{5}\): \[ \left(-8 - 2\sqrt{2}\right) \cdot \frac{4}{5} = \frac{4}{5}(-8) + \frac{4}{5}(-2\sqrt{2}) = -\frac{32}{5} - \frac{8\sqrt{2}}{5} \]

Now let's analyze the result:

• The term \(-\frac{32}{5}\) is a rational number (it is a fraction).
• The term \(-\frac{8\sqrt{2}}{5}\) involves \(\sqrt{2}\), which is irrational. Therefore, this term is also irrational.

When you add a rational number and an irrational number, the result is irrational. Thus, the final result \(-\frac{32}{5} - \frac{8\sqrt{2}}{5}\) is an irrational number.

Therefore, the correct response is: an irrational number.