Question

Which irrational number can be multiplied by –StartRoot 41 EndRoot to get a product that equals 1?
StartRoot StartFraction 1 Over 41 EndFraction EndRoot
Negative StartRoot 41 EndRoot
Negative StartRoot StartFraction 1 Over 41 EndFraction EndRoot
StartRoot 41 EndRoot

Answer 1

To find the irrational number \( x \) that can be multiplied by \( -\sqrt{41} \) to get a product of 1, we can set up the equation:

\[
-\sqrt{41} \cdot x = 1
\]

To solve for \( x \), we divide both sides by \( -\sqrt{41} \):

\[
x = \frac{1}{-\sqrt{41}} = -\frac{1}{\sqrt{41}}
\]

Since the options include square roots and negative roots, we can express \( -\frac{1}{\sqrt{41}} \) in a way that matches one of the options. This can also be rewritten by rationalizing the denominator:

\[
x = -\frac{1}{\sqrt{41}} \cdot \frac{\sqrt{41}}{\sqrt{41}} = -\frac{\sqrt{41}}{41}
\]

However, since none of the options directly show this form, we can select \( -\sqrt{\frac{1}{41}} \) which is equal to \( -\frac{1}{\sqrt{41}} \).

Among the provided options:

1. \( \sqrt{\frac{1}{41}} \)
2. \( -\sqrt{41} \)
3. \( -\sqrt{\frac{1}{41}} \)
4. \( \sqrt{41} \)

The correct choice that matches \( -\frac{1}{\sqrt{41}} \) is:

\[
\text{Negative } \sqrt{\frac{1}{41}} \text{ (which is option 3)}
\]

So the answer is:

\(\mathbf{-\sqrt{\frac{1}{41}}}\)