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

Question

and 45
?(1 point)
Responses

a rational number
a rational number

an irrational number
an irrational number

the number zero
the number zero

an integer

Answer 1

To determine the result of multiplying \(-8 - \sqrt{8}\) and \(45\), we can break it down:

1. \(-8\) is an integer.
2. \(-\sqrt{8}\) is the same as \(-\sqrt{4 \cdot 2} = -2\sqrt{2}\), which is an irrational number.

When you add an integer (-8) and an irrational number (-2√2), the result, \(-8 - 2\sqrt{2}\), is still an irrational number.

Now, when you multiply this entire expression by \(45\):

\[ 45(-8 - 2\sqrt{2}) = 45(-8) + 45(-2\sqrt{2}) \]

This simplifies to:

\[ -360 - 90\sqrt{2} \]

Both terms in this expression contribute to the overall type of the result:

• \(-360\) is a rational number.
• \(-90\sqrt{2}\) is irrational (since \(\sqrt{2}\) is irrational).

When you combine a rational number and an irrational number, the result is generally irrational.

Thus, the final product will be an irrational number.

So the correct response is an irrational number.