Rationalising the Expression
To rationalise the expression \(\frac{8\sqrt{2} + 4\sqrt{3}}{2\sqrt{2} + 2\sqrt{3}}\), we will multiply both the numerator and the denominator by the conjugate of the denominator.
Step 1: Identify the Conjugate
- The denominator is \(2\sqrt{2} + 2\sqrt{3}\).
- The conjugate is \(2\sqrt{2} - 2\sqrt{3}\).
Step 2: Multiply by the Conjugate
- Multiply the numerator and denominator by the conjugate:
\[
\frac{(8\sqrt{2} + 4\sqrt{3})(2\sqrt{2} - 2\sqrt{3})}{(2\sqrt{2} + 2\sqrt{3})(2\sqrt{2} - 2\sqrt{3})}
\]
Step 3: Simplify the Denominator
- The denominator becomes:
\[
(2\sqrt{2})^2 - (2\sqrt{3})^2 = 8 - 12 = -4
\]
Step 4: Expand the Numerator
- The numerator simplifies as follows:
\[
(8\sqrt{2})(2\sqrt{2}) - (8\sqrt{2})(2\sqrt{3}) + (4\sqrt{3})(2\sqrt{2}) - (4\sqrt{3})(2\sqrt{3})
\]
- Calculating each term:
- \(16 \cdot 2 = 32\)
- \(-16\sqrt{6}\)
- \(8\sqrt{6}\)
- \(-12\)
- Therefore, the numerator is:
\[
32 - 12 - 16\sqrt{6} + 8\sqrt{6} = 20 - 8\sqrt{6}
\]
Step 5: Final Expression
- Combining everything, we obtain:
\[
\frac{20 - 8\sqrt{6}}{-4} = -5 + 2\sqrt{6}
\]
Conclusion
- The rationalised form of the expression \(\frac{8\sqrt{2} + 4\sqrt{3}}{2\sqrt{2} + 2\sqrt{3}}\) is:
\[
2\sqrt{6} - 5
\]