The expression given is:

$\frac{1}{\sqrt{10} + \sqrt{6}}$

The instruction is to simplify this using the conjugate.

Step-by-step simplification:

1. Multiply by the conjugate: The conjugate of $\sqrt{10} + \sqrt{6}$ is $\sqrt{10} - \sqrt{6}$. So, multiply the numerator and the denominator by this conjugate: $\frac{1}{\sqrt{10} + \sqrt{6}} \times \frac{\sqrt{10} - \sqrt{6}}{\sqrt{10} - \sqrt{6}} = \frac{\sqrt{10} - \sqrt{6}}{(\sqrt{10})^2 - (\sqrt{6})^2}$

2. Simplify the denominator: The denominator becomes a difference of squares: $(\sqrt{10})^2 - (\sqrt{6})^2 = 10 - 6 = 4$ So, the expression simplifies to: $\frac{\sqrt{10} - \sqrt{6}}{4}$

Thus, the simplified form of the expression is: $\frac{\sqrt{10} - \sqrt{6}}{4}$

For the answer in the box, the conjugate used is $\sqrt{10} - \sqrt{6}$.

Would you like more details on any of the steps, or have any questions?

Relative Questions:

1. What are conjugates, and how do they work in simplification?
2. How is the difference of squares applied in this case?
3. What are other examples of using conjugates to simplify expressions?
4. Can you explain how to multiply expressions involving radicals?
5. How do you deal with complex fractions involving square roots?

Tip:

Always check for conjugates when simplifying expressions with radicals in the denominator—it helps eliminate the square roots effectively.