Understanding the Problem
To rationalise the denominator of the expression \(\sqrt{6}(\sqrt{2}+\sqrt{3})\), we need to eliminate the square root from the denominator.
Step 1: Identify the Denominator
The denominator here is:
- \(D = \sqrt{2} + \sqrt{3}\)
Step 2: Multiply by the Conjugate
To rationalise the denominator, multiply both the numerator and denominator by the conjugate of the denominator:
- Conjugate of \(D\) is \(\sqrt{2} - \sqrt{3}\)
Step 3: Multiply the Expression
Now, we multiply the expression:
\[
\frac{\sqrt{6}(\sqrt{2}+\sqrt{3})}{\sqrt{2}+\sqrt{3}} \times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}
\]
This gives us:
\[
\frac{\sqrt{6}(\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3})}{(\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3})}
\]
Step 4: Simplify the Denominator
Using the difference of squares in the denominator:
- \( (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1 \)
Step 5: Simplify the Numerator
Now, simplify the numerator:
- \(\sqrt{6}((\sqrt{2})^2 - (\sqrt{3})^2) = \sqrt{6}(2 - 3) = \sqrt{6}(-1) = -\sqrt{6}\)
Final Expression
Thus, the expression simplifies to:
\[
\frac{-\sqrt{6}}{-1} = \sqrt{6}
\]
Conclusion
The rationalised form of \(\sqrt{6}(\sqrt{2}+\sqrt{3})\) is:
- \(\sqrt{6}\)