Answer :
Let's rationalize each fraction given:
1. [tex]\(\frac{4}{-3-\sqrt{5}}\)[/tex]:
To rationalize [tex]\(\frac{4}{-3-\sqrt{5}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((-3+\sqrt{5})\)[/tex]:
[tex]\[ \frac{4}{-3-\sqrt{5}} \cdot \frac{-3+\sqrt{5}}{-3+\sqrt{5}} = \frac{4(-3+\sqrt{5})}{(-3-\sqrt{5})(-3+\sqrt{5})} \][/tex]
- Simplify:
[tex]\[ \frac{-12 + 4\sqrt{5}}{9 - (\sqrt{5})^2} = \frac{-12 + 4\sqrt{5}}{4} = -0.763932022500210 \][/tex]
2. [tex]\(-\frac{1}{4+4\sqrt{5}}\)[/tex]:
To rationalize [tex]\(-\frac{1}{4+4\sqrt{5}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((4-4\sqrt{5})\)[/tex]:
[tex]\[ -\frac{1}{4+4\sqrt{5}} \cdot \frac{4-4\sqrt{5}}{4-4\sqrt{5}} = -\frac{4-4\sqrt{5}}{16 - (4\sqrt{5})^2} \][/tex]
- Simplify:
[tex]\[ -\frac{4 - 4\sqrt{5}}{4(-4)} = -0.0772542485937369 \][/tex]
3. [tex]\(\frac{5}{4+\sqrt{5}}\)[/tex]:
To rationalize [tex]\(\frac{5}{4+\sqrt{5}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((4-\sqrt{5})\)[/tex]:
[tex]\[ \frac{5}{4+\sqrt{5}} \cdot \frac{4-\sqrt{5}}{4-\sqrt{5}} = \frac{5(4-\sqrt{5})}{(4+\sqrt{5})(4-\sqrt{5})} \][/tex]
- Simplify:
[tex]\[ \frac{20 - 5\sqrt{5}}{16 - 5} = \frac{20 - 5\sqrt{5}}{11} = 0.801787282954641 \][/tex]
4. [tex]\(\frac{2}{2+3\sqrt{3}}\)[/tex]:
To rationalize [tex]\(\frac{2}{2+3\sqrt{3}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((2-3\sqrt{3})\)[/tex]:
[tex]\[ \frac{2}{2+3\sqrt{3}} \cdot \frac{2-3\sqrt{3}}{2-3\sqrt{3}} = \frac{2(2-3\sqrt{3})}{(2+3\sqrt{3})(2-3\sqrt{3})} \][/tex]
- Simplify:
[tex]\[ \frac{4 - 6\sqrt{3}}{4 - 27} = \frac{4 - 6\sqrt{3}}{-23} = 0.277926297626664 \][/tex]
5. [tex]\(\frac{\sqrt{5}}{4+4\sqrt{3}}\)[/tex]:
To rationalize [tex]\(\frac{\sqrt{5}}{4+4\sqrt{3}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((4-4\sqrt{3})\)[/tex]:
[tex]\[ \frac{\sqrt{5}}{4+4\sqrt{3}} \cdot \frac{4-4\sqrt{3}}{4-4\sqrt{3}} = \frac{\sqrt{5}(4-4\sqrt{3})}{(4+4\sqrt{3})(4-4\sqrt{3})} \][/tex]
- Simplify:
[tex]\[ \frac{4\sqrt{5} - 4\sqrt{15}}{16 - 48} = \frac{4\sqrt{5} - 4\sqrt{15}}{-32} = 0.204614421088453 \][/tex]
6. [tex]\(\frac{4}{5+2\sqrt{3}}\)[/tex]:
To rationalize [tex]\(\frac{4}{5+2\sqrt{3}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((5-2\sqrt{3})\)[/tex]:
[tex]\[ \frac{4}{5+2\sqrt{3}} \cdot \frac{5-2\sqrt{3}}{5-2\sqrt{3}} = \frac{4(5-2\sqrt{3})}{(5+2\sqrt{3})(5-2\sqrt{3})} \][/tex]
- Simplify:
[tex]\[ \frac{20 - 8\sqrt{3}}{25-12} = \frac{20 - 8\sqrt{3}}{13} = 0.472584118419152 \][/tex]
The results are:
1. [tex]\(-0.763932022500210\)[/tex]
2. [tex]\(-0.0772542485937369\)[/tex]
3. [tex]\(0.801787282954641\)[/tex]
4. [tex]\(0.277926297626664\)[/tex]
5. [tex]\(0.204614421088453\)[/tex]
6. [tex]\(0.472584118419152\)[/tex]
1. [tex]\(\frac{4}{-3-\sqrt{5}}\)[/tex]:
To rationalize [tex]\(\frac{4}{-3-\sqrt{5}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((-3+\sqrt{5})\)[/tex]:
[tex]\[ \frac{4}{-3-\sqrt{5}} \cdot \frac{-3+\sqrt{5}}{-3+\sqrt{5}} = \frac{4(-3+\sqrt{5})}{(-3-\sqrt{5})(-3+\sqrt{5})} \][/tex]
- Simplify:
[tex]\[ \frac{-12 + 4\sqrt{5}}{9 - (\sqrt{5})^2} = \frac{-12 + 4\sqrt{5}}{4} = -0.763932022500210 \][/tex]
2. [tex]\(-\frac{1}{4+4\sqrt{5}}\)[/tex]:
To rationalize [tex]\(-\frac{1}{4+4\sqrt{5}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((4-4\sqrt{5})\)[/tex]:
[tex]\[ -\frac{1}{4+4\sqrt{5}} \cdot \frac{4-4\sqrt{5}}{4-4\sqrt{5}} = -\frac{4-4\sqrt{5}}{16 - (4\sqrt{5})^2} \][/tex]
- Simplify:
[tex]\[ -\frac{4 - 4\sqrt{5}}{4(-4)} = -0.0772542485937369 \][/tex]
3. [tex]\(\frac{5}{4+\sqrt{5}}\)[/tex]:
To rationalize [tex]\(\frac{5}{4+\sqrt{5}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((4-\sqrt{5})\)[/tex]:
[tex]\[ \frac{5}{4+\sqrt{5}} \cdot \frac{4-\sqrt{5}}{4-\sqrt{5}} = \frac{5(4-\sqrt{5})}{(4+\sqrt{5})(4-\sqrt{5})} \][/tex]
- Simplify:
[tex]\[ \frac{20 - 5\sqrt{5}}{16 - 5} = \frac{20 - 5\sqrt{5}}{11} = 0.801787282954641 \][/tex]
4. [tex]\(\frac{2}{2+3\sqrt{3}}\)[/tex]:
To rationalize [tex]\(\frac{2}{2+3\sqrt{3}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((2-3\sqrt{3})\)[/tex]:
[tex]\[ \frac{2}{2+3\sqrt{3}} \cdot \frac{2-3\sqrt{3}}{2-3\sqrt{3}} = \frac{2(2-3\sqrt{3})}{(2+3\sqrt{3})(2-3\sqrt{3})} \][/tex]
- Simplify:
[tex]\[ \frac{4 - 6\sqrt{3}}{4 - 27} = \frac{4 - 6\sqrt{3}}{-23} = 0.277926297626664 \][/tex]
5. [tex]\(\frac{\sqrt{5}}{4+4\sqrt{3}}\)[/tex]:
To rationalize [tex]\(\frac{\sqrt{5}}{4+4\sqrt{3}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((4-4\sqrt{3})\)[/tex]:
[tex]\[ \frac{\sqrt{5}}{4+4\sqrt{3}} \cdot \frac{4-4\sqrt{3}}{4-4\sqrt{3}} = \frac{\sqrt{5}(4-4\sqrt{3})}{(4+4\sqrt{3})(4-4\sqrt{3})} \][/tex]
- Simplify:
[tex]\[ \frac{4\sqrt{5} - 4\sqrt{15}}{16 - 48} = \frac{4\sqrt{5} - 4\sqrt{15}}{-32} = 0.204614421088453 \][/tex]
6. [tex]\(\frac{4}{5+2\sqrt{3}}\)[/tex]:
To rationalize [tex]\(\frac{4}{5+2\sqrt{3}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((5-2\sqrt{3})\)[/tex]:
[tex]\[ \frac{4}{5+2\sqrt{3}} \cdot \frac{5-2\sqrt{3}}{5-2\sqrt{3}} = \frac{4(5-2\sqrt{3})}{(5+2\sqrt{3})(5-2\sqrt{3})} \][/tex]
- Simplify:
[tex]\[ \frac{20 - 8\sqrt{3}}{25-12} = \frac{20 - 8\sqrt{3}}{13} = 0.472584118419152 \][/tex]
The results are:
1. [tex]\(-0.763932022500210\)[/tex]
2. [tex]\(-0.0772542485937369\)[/tex]
3. [tex]\(0.801787282954641\)[/tex]
4. [tex]\(0.277926297626664\)[/tex]
5. [tex]\(0.204614421088453\)[/tex]
6. [tex]\(0.472584118419152\)[/tex]
