Réitigh frac{sqrt{18}-sqrt{12}}{sqrt{50}-sqrt{48}}

$$\frac { \sqrt { 18 } - \sqrt { 12 } } { \sqrt { 50 } - \sqrt { 48 } }$$

Luacháil

$$\sqrt{6}+3\approx 5.449489743$$

Céimeanna Réitigh

Fachtóirigh 18=3^{2}\times 2. Athscríobh fréamh cearnach an toraidh \sqrt{3^{2}\times 2} mar thoradh na bhfréamhacha cearnacha \sqrt{3^{2}}\sqrt{2}. Tóg fréamh chearnach 3^{2}.

$$\frac{3\sqrt{2}-\sqrt{12}}{\sqrt{50}-\sqrt{48}}$$

Fachtóirigh 12=2^{2}\times 3. Athscríobh fréamh cearnach an toraidh \sqrt{2^{2}\times 3} mar thoradh na bhfréamhacha cearnacha \sqrt{2^{2}}\sqrt{3}. Tóg fréamh chearnach 2^{2}.

$$\frac{3\sqrt{2}-2\sqrt{3}}{\sqrt{50}-\sqrt{48}}$$

Fachtóirigh 50=5^{2}\times 2. Athscríobh fréamh cearnach an toraidh \sqrt{5^{2}\times 2} mar thoradh na bhfréamhacha cearnacha \sqrt{5^{2}}\sqrt{2}. Tóg fréamh chearnach 5^{2}.

$$\frac{3\sqrt{2}-2\sqrt{3}}{5\sqrt{2}-\sqrt{48}}$$

Fachtóirigh 48=4^{2}\times 3. Athscríobh fréamh cearnach an toraidh \sqrt{4^{2}\times 3} mar thoradh na bhfréamhacha cearnacha \sqrt{4^{2}}\sqrt{3}. Tóg fréamh chearnach 4^{2}.

$$\frac{3\sqrt{2}-2\sqrt{3}}{5\sqrt{2}-4\sqrt{3}}$$

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi 5\sqrt{2}+4\sqrt{3} chun ainmneoir \frac{3\sqrt{2}-2\sqrt{3}}{5\sqrt{2}-4\sqrt{3}} a thiontú in uimhir chóimheasta.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{\left(5\sqrt{2}-4\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}$$

Mar shampla \left(5\sqrt{2}-4\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right). Is féidir iolrúchán a athrú ó bhonn go dtí difríocht na gcearnóg ag úsáid na rialach seo: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{\left(5\sqrt{2}\right)^{2}-\left(-4\sqrt{3}\right)^{2}}$$

Fairsingigh \left(5\sqrt{2}\right)^{2}

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{5^{2}\left(\sqrt{2}\right)^{2}-\left(-4\sqrt{3}\right)^{2}}$$

Ríomh cumhacht 5 de 2 agus faigh 25.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{25\left(\sqrt{2}\right)^{2}-\left(-4\sqrt{3}\right)^{2}}$$

Is é 2 uimhir chearnach \sqrt{2}.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{25\times 2-\left(-4\sqrt{3}\right)^{2}}$$

Méadaigh 25 agus 2 chun 50 a fháil.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-\left(-4\sqrt{3}\right)^{2}}$$

Fairsingigh \left(-4\sqrt{3}\right)^{2}

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-\left(-4\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Ríomh cumhacht -4 de 2 agus faigh 16.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-16\left(\sqrt{3}\right)^{2}}$$

Is é 3 uimhir chearnach \sqrt{3}.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-16\times 3}$$

Méadaigh 16 agus 3 chun 48 a fháil.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-48}$$

Dealaigh 48 ó 50 chun 2 a fháil.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{2}$$

Cuir an t-airí dáileacháin i bhfeidhm trí gach téarma de 3\sqrt{2}-2\sqrt{3} a iolrú faoi gach téarma de 5\sqrt{2}+4\sqrt{3}.

$$\frac{15\left(\sqrt{2}\right)^{2}+12\sqrt{3}\sqrt{2}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

Is é 2 uimhir chearnach \sqrt{2}.

$$\frac{15\times 2+12\sqrt{3}\sqrt{2}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

Méadaigh 15 agus 2 chun 30 a fháil.

$$\frac{30+12\sqrt{3}\sqrt{2}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

Iolraigh na huimhreacha faoin bhfréamh cearnach chun \sqrt{3} agus \sqrt{2} a iolrú.

$$\frac{30+12\sqrt{6}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

Iolraigh na huimhreacha faoin bhfréamh cearnach chun \sqrt{3} agus \sqrt{2} a iolrú.

$$\frac{30+12\sqrt{6}-10\sqrt{6}-8\left(\sqrt{3}\right)^{2}}{2}$$

Comhcheangail 12\sqrt{6} agus -10\sqrt{6} chun 2\sqrt{6} a fháil.

$$\frac{30+2\sqrt{6}-8\left(\sqrt{3}\right)^{2}}{2}$$

Is é 3 uimhir chearnach \sqrt{3}.

$$\frac{30+2\sqrt{6}-8\times 3}{2}$$

Méadaigh -8 agus 3 chun -24 a fháil.

$$\frac{30+2\sqrt{6}-24}{2}$$

Dealaigh 24 ó 30 chun 6 a fháil.

$$\frac{6+2\sqrt{6}}{2}$$

Roinn 6+2\sqrt{6} faoi 2 chun 3+\sqrt{6} a fháil.

$$3+\sqrt{6}$$