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Q4: Rationalize the following expression:

[tex]\[ \frac{3 + 2 \sqrt{3}}{3 - 2 \sqrt{2}} \][/tex]


Sagot :

To rationalize the expression [tex]\(\frac{3 + 2\sqrt{3}}{3 - 2\sqrt{2}}\)[/tex], we follow these steps:

1. Identify the Conjugate:
For the denominator [tex]\(3 - 2\sqrt{2}\)[/tex], the conjugate is [tex]\(3 + 2\sqrt{2}\)[/tex].

2. Multiply the Numerator and Denominator by the Conjugate:
Multiply both the numerator and the denominator by the conjugate of the denominator to remove the irrational part from the denominator.
[tex]\[ \frac{3 + 2\sqrt{3}}{3 - 2\sqrt{2}} \times \frac{3 + 2\sqrt{2}}{3 + 2\sqrt{2}} \][/tex]

3. Expand the Numerator:
[tex]\[ (3 + 2\sqrt{3})(3 + 2\sqrt{2}) \][/tex]
Using the distributive property:
[tex]\[ = 3 \cdot 3 + 3 \cdot 2\sqrt{2} + 2\sqrt{3} \cdot 3 + 2\sqrt{3} \cdot 2\sqrt{2} \][/tex]
Simplify the terms:
[tex]\[ = 9 + 6\sqrt{2} + 6\sqrt{3} + 4\sqrt{6} \][/tex]

4. Simplify the Denominator:
[tex]\[ (3 - 2\sqrt{2})(3 + 2\sqrt{2}) \][/tex]
This is a difference of squares:
[tex]\[ = 3^2 - (2\sqrt{2})^2 \][/tex]
Simplify the terms:
[tex]\[ = 9 - 4 \cdot 2 \][/tex]
[tex]\[ = 9 - 8 = 1 \][/tex]

5. Write the Rationalized Expression:
The numerator simplifies to approximately 37.67554519078455 and the denominator simplifies to approximately 1. Therefore, the rationalized expression is:
[tex]\[ \frac{3 + 2\sqrt{3})(3 + 2\sqrt{2})}{(3 - 2\sqrt{2})(3 + 2\sqrt{2})} = 37.67554519078455 \][/tex]
The final answer is very close to:
[tex]\[ \boxed{37.67554519078461} \][/tex]