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 that can be multiplied by \(-\sqrt{41}\) to get a product of 1, we need to solve the equation:

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

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

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

Now, we can write this in a more recognizable form by simplifying the fraction. To eliminate the square root from the denominator, we can multiply the numerator and the denominator by \(\sqrt{41}\):

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

Now, let's check the provided options:

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

None of these options are exactly the same as \(-\frac{\sqrt{41}}{41}\), but \( -\sqrt{\frac{1}{41}} \) simplifies to:

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

This matches our derived solution of \( -\frac{1}{\sqrt{41}} \), leading us to conclude that the correct choice from the options provided is:

**Option 3: \(-\sqrt{\frac{1}{41}}\)**.