Terdapat beberapa bentuk akar. Hasil merasionalkan bentuk-bentuk akar tersebut adalah sebagai berikut:
- [tex]\frac{10}{3+\sqrt{3}}=\bf5-\frac{5}{3}\sqrt{3}[/tex]
- [tex]\frac{\sqrt{10}}{4-\sqrt{2}}=\bf\frac{2}{7}\sqrt{10}+\frac{1}{7}\sqrt{5}[/tex]
- [tex]\frac{2\sqrt{5}}{5+\sqrt{3}}=\bf\frac{5}{11}\sqrt{5}-\frac{1}{11}\sqrt{15}[/tex]
- [tex]\frac{4\sqrt{6}}{\sqrt{3}-\sqrt{2}}=\bf12\sqrt{2}+8\sqrt{3}[/tex]
- [tex]\frac{4+\sqrt{2}}{2\sqrt{3}+3\sqrt{3}}=\bf\frac{4}{15}\sqrt{3}+\frac{1}{15}\sqrt{6}[/tex]
Penjelasan dengan langkah-langkah
Diketahui:
- [tex]\frac{10}{3+\sqrt{3}}[/tex]
- [tex]\frac{\sqrt{10}}{4-\sqrt{2}}[/tex]
- [tex]\frac{2\sqrt{5}}{5+\sqrt{3}}[/tex]
- [tex]\frac{4\sqrt{6}}{\sqrt{3}-\sqrt{2}}[/tex]
- [tex]\frac{4+\sqrt{2}}{2\sqrt{3}+3\sqrt{3}}[/tex]
Ditanya: hasil perasionalan bentuk akar
Jawab:
Untuk nomor 1:
3-√3
[tex]\frac{10}{3+\sqrt{3}}=\frac{10}{3+\sqrt{3}}\cdot\frac{3-\sqrt{3}}{3-\sqrt{3}}\\=\frac{10(3-\sqrt{3})}{3^2-(\sqrt{3})^2}\\=\frac{10(3-\sqrt{3})}{9-3}\\=\frac{10(3-\sqrt{3})}{6}\\=\frac{5(3-\sqrt{3})}{3}\\=\frac{15-5\sqrt{3}}{3}\\=5-\frac{5}{3}\sqrt{3}[/tex]
Untuk nomor 2:
4+√2
[tex]\frac{\sqrt{10}}{4-\sqrt{2}}=\frac{\sqrt{10}}{4-\sqrt{2}}\cdot\frac{4+\sqrt{2}}{4+\sqrt{2}}\\=\frac{4\sqrt{10}+\sqrt{20}}{4^2-(\sqrt{2})^2}\\=\frac{4\sqrt{10}+2\sqrt{5}}{16-2}\\=\frac{4\sqrt{10}+2\sqrt{5}}{14}\\=\frac{2\sqrt{10}+\sqrt{5}}{7}\\=\frac{2}{7}\sqrt{10}+\frac{1}{7}\sqrt{5}[/tex]
Untuk nomor 3:
5-√3
[tex]\frac{2\sqrt{5}}{5+\sqrt{3}}=\frac{2\sqrt{5}}{5+\sqrt{3}}\cdot\frac{5-\sqrt{3}}{5-\sqrt{3}}\\=\frac{2\sqrt{5}(5-\sqrt{3})}{5^2-(\sqrt{3})^2}\\=\frac{2\sqrt{5}(5-\sqrt{3})}{25-3}\\=\frac{2\sqrt{5}(5-\sqrt{3})}{22}\\=\frac{\sqrt{5}(5-\sqrt{3})}{11}\\=\frac{5\sqrt{5}-\sqrt{15}}{11}\\=\frac{5}{11}\sqrt{5}-\frac{1}{11}\sqrt{15}[/tex]
Untuk nomor 4:
√3+√2
[tex]\frac{4\sqrt{6}}{\sqrt{3}-\sqrt{2}}=\frac{4\sqrt{6}}{\sqrt{3}-\sqrt{2}}\cdot\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}\\=\frac{4\sqrt{6}(\sqrt{3}+\sqrt{2})}{(\sqrt{3})^2-(\sqrt{2})^2}\\=\frac{4\sqrt{18}+4\sqrt{12}}{3-2}\\=\frac{12\sqrt{2}+8\sqrt{3}}{1}\\=12\sqrt{2}+8\sqrt{3}[/tex]
Untuk nomor 5:
- Bentuk sederhana penyebut
2√2+3√3 = 5√3
[tex]\frac{4+\sqrt{2}}{2\sqrt{3}+3\sqrt{3}}=\frac{4+\sqrt{2}}{5\sqrt{3}}\\=\frac{4+\sqrt{2}}{5\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}\\=\frac{4\sqrt{3}+\sqrt{6}}{5\cdot3}\\=\frac{4\sqrt{3}+\sqrt{6}}{15}\\=\frac{4}{15}\sqrt{3}+\frac{1}{15}\sqrt{6}[/tex]
Pelajari lebih lanjut
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