Simplify: 3/(3+√5) + (3√5)/(3-√5)

Question

ScanSolve AI Tutor · Accepted Answer

Ili kurahisisha usemi huu, tutaweka sawa denominator ya kila sehemu na kisha kuzijumlisha.

Rahisisha sehemu ya kwanza kwa kuondoa mizizi kwenye denominator.
Tunazidisha namba na denominator kwa kiambishi cha $3+\sqrt{5}$ , ambacho ni $3-\sqrt{5}$ .
$\frac{3}{3+\sqrt{5}} = \frac{3}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}}$ $= \frac{3(3-\sqrt{5})}{3^2 - (\sqrt{5})^2}$ $= \frac{9-3\sqrt{5}}{9-5}$ $= \frac{9-3\sqrt{5}}{4}$
Rahisisha sehemu ya pili kwa kuondoa mizizi kwenye denominator.
Tunazidisha namba na denominator kwa kiambishi cha $3-\sqrt{5}$ , ambacho ni $3+\sqrt{5}$ .
$\frac{3\sqrt{5}}{3-\sqrt{5}} = \frac{3\sqrt{5}}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}}$ $= \frac{3\sqrt{5}(3+\sqrt{5})}{3^2 - (\sqrt{5})^2}$ $= \frac{3\sqrt{5} \cdot 3 + 3\sqrt{5} \cdot \sqrt{5}}{9-5}$ $= \frac{9\sqrt{5} + 3 \cdot 5}{4}$ $= \frac{15+9\sqrt{5}}{4}$
Jumlisha sehemu zote mbili.
Sasa tuna denominator sawa, tunaweza kujumlisha namba.
$\frac{9-3\sqrt{5}}{4} + \frac{15+9\sqrt{5}}{4} = \frac{(9-3\sqrt{5}) + (15+9\sqrt{5})}{4}$ $= \frac{9+15 - 3\sqrt{5} + 9\sqrt{5}}{4}$ $= \frac{24 + 6\sqrt{5}}{4}$
Rahisisha jibu la mwisho.
Gawanya kila namba kwa 4.
$= \frac{24}{4} + \frac{6\sqrt{5}}{4}$ $= 6 + \frac{3\sqrt{5}}{2}$
Jibu la mwisho ni:
$6 + \frac{3\sqrt{5}}{2}$

Answer