Issolvi frac{isqrt{2}-5}{i+sqrt{2}}

$$\frac { i \sqrt { 2 } - 5 } { i + \sqrt { 2 } }$$

Evalwa

$$-\frac{4\sqrt{2}}{3}+\frac{7}{3}i\approx -1.885618083+2.333333333i$$

Passi tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{i\sqrt{2}-5}{i+\sqrt{2}} billi timmultiplika in-numeratur u d-denominatur mill-i-\sqrt{2}.

$$\frac{\left(i\sqrt{2}-5\right)\left(i-\sqrt{2}\right)}{\left(i+\sqrt{2}\right)\left(i-\sqrt{2}\right)}$$

Ikkunsidra li \left(i+\sqrt{2}\right)\left(i-\sqrt{2}\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(i\sqrt{2}-5\right)\left(i-\sqrt{2}\right)}{i^{2}-\left(\sqrt{2}\right)^{2}}$$

Ikkwadra i. Ikkwadra \sqrt{2}.

$$\frac{\left(i\sqrt{2}-5\right)\left(i-\sqrt{2}\right)}{-1-2}$$

Naqqas 2 minn -1 biex tikseb -3.

$$\frac{\left(i\sqrt{2}-5\right)\left(i-\sqrt{2}\right)}{-3}$$

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' i\sqrt{2}-5 b'kull terminu ta' i-\sqrt{2}.

$$\frac{-\sqrt{2}-i\left(\sqrt{2}\right)^{2}-5i+5\sqrt{2}}{-3}$$

Il-kwadrat ta' \sqrt{2} huwa 2.

$$\frac{-\sqrt{2}-i\times 2-5i+5\sqrt{2}}{-3}$$

Immultiplika -i u 2 biex tikseb -2i.

$$\frac{-\sqrt{2}-2i-5i+5\sqrt{2}}{-3}$$

Naqqas 5i minn -2i biex tikseb -7i.

$$\frac{-\sqrt{2}-7i+5\sqrt{2}}{-3}$$

Ikkombina -\sqrt{2} u 5\sqrt{2} biex tikseb 4\sqrt{2}.

$$\frac{4\sqrt{2}-7i}{-3}$$

Immultiplika kemm in-numeratur u kif ukoll id-denominatur b’-1.

$$\frac{-4\sqrt{2}+7i}{3}$$

Parti Reali

$$-\frac{4 \sqrt{2}}{3} = -1.885618083164127$$