Question

Rasonalkan bentuk pecahan berikut!
[tex] \frac{ \sqrt{3} + \sqrt{2} + \sqrt{5} }{ \sqrt{3} - \sqrt{2} } \\ [/tex]
Dimana, menghasilkan bentuk
[tex] \rm \: p + 2 \sqrt{q} + \sqrt{r} + \sqrt{s} [/tex]
Tentukanlah hasil dari p + q - r + s ! ​

Answer 1

Verified answer

[tex] \frac{ \sqrt{3} + \sqrt{2} + \sqrt{5} }{ \sqrt{3} - \sqrt{2} } \\ [/tex]

Penyelesaian :

[tex] = \frac{( \sqrt{3} + \sqrt{2} + \sqrt{5} ) \times ( \sqrt{3} + \sqrt{2}) }{( \sqrt{3} - \sqrt{2}) \times ( \sqrt{3} + \sqrt{2} } \\ [/tex]

[tex] = \frac{( \sqrt{3} + \sqrt{2} + \sqrt{5} ) \times ( \sqrt{3} + \sqrt{2}) }{( { \sqrt{3}} {)}^{2} - ( \sqrt{2} {)}^{2} } \\ [/tex]

[tex] = \frac{( \sqrt{3} + \sqrt{2} + \sqrt{5}) \times ( \sqrt{ 3} + \sqrt{2}) }{3 - 2} \\ [/tex]

[tex] = \frac{( \sqrt{3} + \sqrt{2} + \sqrt{5}) \times ( \sqrt{3} + \sqrt{2}) }{1} \\ [/tex]

[tex] = ( \sqrt{3} + \sqrt{2} + \sqrt{5}) \times ( \sqrt{3} + \sqrt{2)} [/tex]

[tex] = 3 + \sqrt{6} + \sqrt{6} + 2 + \sqrt{15} + \sqrt{10} [/tex]

[tex] = 5 + \sqrt{6} + \sqrt{6} + + \sqrt{15} + \sqrt{10} [/tex]

[tex] = 5 + 2 \sqrt{6} + \sqrt{15} + \sqrt{10} [/tex]

makaa nilai :

p = 5

q = 6

r = 15

s = 10

p + q - r + s :

5 + 6 - 15 + 10

= 11 - 15 + 10

= -4 + 10

= 6

Answer 2

p + q – r + s = 6.
 

Pembahasan

[tex]\dfrac{\sqrt{3}+\sqrt{2}+\sqrt{5}}{\sqrt{3}-\sqrt{2}}[/tex]

[tex]=\dfrac{\sqrt{3}+\sqrt{2}+\sqrt{5}}{\sqrt{3}-\sqrt{2}}\times\dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}[/tex]

[tex]=\dfrac{\left(\sqrt{3}+\sqrt{2}\right)^2+\sqrt{5}\left(\sqrt{3}+\sqrt{2}\right)}{\left(\sqrt{3}\right)^2-\left(\sqrt{2}\right)^2}[/tex]

[tex]=\dfrac{\left(\sqrt{3}\right)^2+2\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^2+\sqrt{5}\sqrt{3}+\sqrt{5}\sqrt{2}}{3-2}[/tex]

[tex]=\dfrac{3+2\sqrt{6}+2+\sqrt{15}+\sqrt{10}}{1}[/tex]

[tex]={\bf5}+2\sqrt{\bf6}+\sqrt{\bf15}+\sqrt{\bf10}[/tex]

[tex]=\rm p+2\sqrt{q}+\sqrt{r\ }\ \,+\sqrt{s\ }[/tex]

Maka:
p = 5, q = 6, r = 15, s = 10

Sehingga:
p + q – r + s = 5 + 6 – 15 + 10 = 5 + 6 – 5 = 6.