Question

Find a and b, if 4+√5/4-√5
= a + bV5​

Answer 1

Answer:

$a = \frac{21}{11}\: and \: b= \frac{8}{11}$

Step-by-step explanation:

$a+b\sqrt{5}\\=\frac{4+\sqrt{5}}{4-\sqrt{5}}\\=\frac{(4+\sqrt{5})(4+\sqrt{5}}{(4-\sqrt{5})(4+\sqrt{5}}\\=\frac{(4+\sqrt{5})^{2}}{4^{2}-(\sqrt{5})^{2}}\\=\frac{4^{2}+2\times 4\times \sqrt{5}+(\sqrt{5})^{2}}{16-5}\\=\frac{16+8\sqrt{5}+5}{11}\\=\frac{21+8\sqrt{5}}{11}\\=\frac{21}{11}+\frac{8}{11} \sqrt{5}$

$a+b\sqrt{5}= \frac{21}{11}+\frac{8}{11} \sqrt{5}$

Therefore,.

$a = \frac{21}{11}\: and \: b= \frac{8}{11}$

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